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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Generalized Balloons and Chinese Postman
Problems in Regular Graphs
Suil O and Douglas B. West
Department of MathematicsUniversity of Illinois at
Urbana-Champaign
2009 SIAM Annual Meeting
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Table of Contents
Motivation and Questions
Previous Results using Balloons
Smallest Matching Number and Edge-Connectivity
Chinese Postman Problem
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Motivation and Questions
In 1891, Petersen proved that every cubic graph without
cut-edgeshas a perfect matching.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Motivation and Questions
In 1891, Petersen proved that every cubic graph without
cut-edgeshas a perfect matching.If there are cut-edges in a cubic
graph, then what happens?
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Motivation and Questions
In 1891, Petersen proved that every cubic graph without
cut-edgeshas a perfect matching.If there are cut-edges in a cubic
graph, then what happens?More generally, we consider connected (2r
+ 1)-regular graphs.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Motivation and Questions
In 1891, Petersen proved that every cubic graph without
cut-edgeshas a perfect matching.If there are cut-edges in a cubic
graph, then what happens?More generally, we consider connected (2r
+ 1)-regular graphs.
1) How many cut-edges can a connected (2r + 1)-regular graphwith
n vertices have?
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Motivation and Questions
In 1891, Petersen proved that every cubic graph without
cut-edgeshas a perfect matching.If there are cut-edges in a cubic
graph, then what happens?More generally, we consider connected (2r
+ 1)-regular graphs.
1) How many cut-edges can a connected (2r + 1)-regular graphwith
n vertices have?2) How small can the matching number α′(G ) be in a
connected(2r + 1)-regular graph with n vertices?
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Motivation and Questions
In 1891, Petersen proved that every cubic graph without
cut-edgeshas a perfect matching.If there are cut-edges in a cubic
graph, then what happens?More generally, we consider connected (2r
+ 1)-regular graphs.
1) How many cut-edges can a connected (2r + 1)-regular graphwith
n vertices have?2) How small can the matching number α′(G ) be in a
connected(2r + 1)-regular graph with n vertices?3) Can we
characterize when equality holds? (i.e. when is thematching number
minimized?)
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Motivation and Questions
In 1891, Petersen proved that every cubic graph without
cut-edgeshas a perfect matching.If there are cut-edges in a cubic
graph, then what happens?More generally, we consider connected (2r
+ 1)-regular graphs.
1) How many cut-edges can a connected (2r + 1)-regular graphwith
n vertices have?2) How small can the matching number α′(G ) be in a
connected(2r + 1)-regular graph with n vertices?3) Can we
characterize when equality holds? (i.e. when is thematching number
minimized?)4) Are there other applications of balloons?
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Definitions
◮ Graphs in which every vertex has degree 3 are cubic
graphs.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Definitions
◮ Graphs in which every vertex has degree 3 are cubic
graphs.
◮ The matching number α′(G ) of a graph is the maximum sizeof a
matching in it.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Definitions
◮ Graphs in which every vertex has degree 3 are cubic
graphs.
◮ The matching number α′(G ) of a graph is the maximum sizeof a
matching in it.
◮ Let c(G ) denote the number of cut-edges in a graph G .
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Definitions
◮ Graphs in which every vertex has degree 3 are cubic
graphs.
◮ The matching number α′(G ) of a graph is the maximum sizeof a
matching in it.
◮ Let c(G ) denote the number of cut-edges in a graph G .
◮ Let b(G ) denote the number of balloons in a graph G .
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Tools - Balloons and Br
B1 a cubic graph with two balloons
◮ A balloon in a graph G is a maximal 2-edge-connectedsubgraph
of G that is incident to exactly one cut-edge of G .
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Tools - Balloons and Br
B1 a cubic graph with two balloons
◮ A balloon in a graph G is a maximal 2-edge-connectedsubgraph
of G that is incident to exactly one cut-edge of G .
◮ Br : the unique graph with 2r + 3 vertices having 2r +
2vertices of degree 2r + 1 and one vertex of degree 2r .(Br is the
complement of P3 + rK2)
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Tools - Balloons and Br
B1 a cubic graph with two balloons
◮ A balloon in a graph G is a maximal 2-edge-connectedsubgraph
of G that is incident to exactly one cut-edge of G .
◮ Br : the unique graph with 2r + 3 vertices having 2r +
2vertices of degree 2r + 1 and one vertex of degree 2r .(Br is the
complement of P3 + rK2)
◮ Br is the smallest possible balloon in a (2r + 1)-regular
graph.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Previous Results
Fn : the family of connected (2r + 1)-regular graphs with
nvertices.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Previous Results
Fn : the family of connected (2r + 1)-regular graphs with
nvertices.
Theorem (O and West 2009+)
If G ∈ Fn, then c(G ) ≤r(n−2)−22r2+2r−1
− 1 cut-edges, which reduces ton−73 for cubic graphs. Equality
holds infinitely often.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Previous Results
Fn : the family of connected (2r + 1)-regular graphs with
nvertices.
Theorem (O and West 2009+)
If G ∈ Fn, then c(G ) ≤r(n−2)−22r2+2r−1
− 1 cut-edges, which reduces ton−73 for cubic graphs. Equality
holds infinitely often.
Theorem (Henning and Yeo 2007)
If G ∈ Fn, then α′(G ) ≥ n2 −
r2
(2r−1)n+2(2r+1)(2r2+2r−1)
.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Construction - Tr and Hr
a graph in T1 a graph in H1
◮ Tr : the family of trees such that every non-leaf vertex
hasdegree 2r + 1 and all the leaves have the same color in aproper
2-coloring.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Construction - Tr and Hr
a graph in T1 a graph in H1
◮ Tr : the family of trees such that every non-leaf vertex
hasdegree 2r + 1 and all the leaves have the same color in aproper
2-coloring.
◮ Hr : the family of (2r + 1)-regular graphs obtained from
treesin Tr by identifying each leaf of such a tree with the vertex
ofdegree 2r in a copy of Br .
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Properties of graphs in Hr
Proposition (O and West 2009+)
Let G be an n-vertex graph in Hr .
◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Properties of graphs in Hr
Proposition (O and West 2009+)
Let G be an n-vertex graph in Hr .
◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)
◮ b(G ) = (2r−1)n+24r2+4r−2
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Properties of graphs in Hr
Proposition (O and West 2009+)
Let G be an n-vertex graph in Hr .
◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)
◮ b(G ) = (2r−1)n+24r2+4r−2
◮ α′(G ) = 12n −r2
(2r−1)n+2r(4r3+6r2−1)
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Properties of graphs in Hr
Proposition (O and West 2009+)
Let G be an n-vertex graph in Hr .
◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)
◮ b(G ) = (2r−1)n+24r2+4r−2
◮ α′(G ) = 12n −r2
(2r−1)n+2r(4r3+6r2−1)
◮ c(G ) = r(n−2)−22r2+2r−1
− 1
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Properties of graphs in Hr
Proposition (O and West 2009+)
Let G be an n-vertex graph in Hr .
◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)
◮ b(G ) = (2r−1)n+24r2+4r−2
◮ α′(G ) = 12n −r2
(2r−1)n+2r(4r3+6r2−1)
◮ c(G ) = r(n−2)−22r2+2r−1
− 1
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Properties of graphs in Hr
Proposition (O and West 2009+)
Let G be an n-vertex graph in Hr .
◮ n ≡ 4(r + 1)2 mod (8r3 + 12r2 − 2)
◮ b(G ) = (2r−1)n+24r2+4r−2
◮ α′(G ) = 12n −r2
(2r−1)n+2r(4r3+6r2−1)
◮ c(G ) = r(n−2)−22r2+2r−1
− 1
Theorem (O and West 2009+)
When n is as above, G has the smallest matching number over
allconnected (2r + 1)-regular graphs if and only if G is in Hr
.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Balloons and Total Domination
◮ A subset S is a dominating set in a graph G if every
vertexoutside S has a neighbor in S .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Balloons and Total Domination
◮ A subset S is a dominating set in a graph G if every
vertexoutside S has a neighbor in S .
◮ A subset S is a total dominating set in a graph G if
everyvertex in V (G ) has a neighbor in S . The total
dominationnumber of G , denoted γt(G ), is the least size of such a
set.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Balloons and Total Domination
◮ A subset S is a dominating set in a graph G if every
vertexoutside S has a neighbor in S .
◮ A subset S is a total dominating set in a graph G if
everyvertex in V (G ) has a neighbor in S . The total
dominationnumber of G , denoted γt(G ), is the least size of such a
set.
Theorem (Henning, Kang, Shan, and Yeo 2008)
When G is regular with degree at least 3, γt(G ) ≤ α′(G ).
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Balloons and Total Domination
◮ A subset S is a dominating set in a graph G if every
vertexoutside S has a neighbor in S .
◮ A subset S is a total dominating set in a graph G if
everyvertex in V (G ) has a neighbor in S . The total
dominationnumber of G , denoted γt(G ), is the least size of such a
set.
Theorem (Henning, Kang, Shan, and Yeo 2008)
When G is regular with degree at least 3, γt(G ) ≤ α′(G ).
Theorem (O and West 2009+)
If G is a connected cubic graph, then γt(G ) ≤ α′(G ) − b(G
)/6,
unless b(G ) = 3 and there is only one vertex outside the
balloons.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
New Questions
◮ What is the smallest matching number for the family
oft-edge-connected (2r + 1)-regular graphs with n vertices?(Also
for 2r -regular graphs.)
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
New Questions
◮ What is the smallest matching number for the family
oft-edge-connected (2r + 1)-regular graphs with n vertices?(Also
for 2r -regular graphs.)
◮ What is the best upper bound for the length of Chinesepostman
tours in t-edge-connected (2r + 1)-regular graphswith n
vertices?
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Smallest Matching Number and Edge-Connectivity
Theorem (O and West 2009+)
If G is a (2t + 1)-edge-connected (2r + 1)-regular graph with
nvertices, then α′(G ) ≥ n2 − (
r−t2(r+1)2+t
)n2 , and this is sharp for t ≥ 1.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Smallest Matching Number and Edge-Connectivity
Theorem (O and West 2009+)
If G is a (2t + 1)-edge-connected (2r + 1)-regular graph with
nvertices, then α′(G ) ≥ n2 − (
r−t2(r+1)2+t
)n2 , and this is sharp for t ≥ 1.
Proof. Let S be a set with maximum deficiency. Thus,α′(G ) = 12
(n − def(S)), where def(S) = o(G − S) − |S |.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Smallest Matching Number and Edge-Connectivity
Theorem (O and West 2009+)
If G is a (2t + 1)-edge-connected (2r + 1)-regular graph with
nvertices, then α′(G ) ≥ n2 − (
r−t2(r+1)2+t
)n2 , and this is sharp for t ≥ 1.
Proof. Let S be a set with maximum deficiency. Thus,α′(G ) = 12
(n − def(S)), where def(S) = o(G − S) − |S |.
Let ci count the odd components of G − S having i edges to S
.Let c = c(2t+1) + ... + c(2r−1), and let c
′ = o(G − S) − c .
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Smallest Matching Number and Edge-Connectivity
Theorem (O and West 2009+)
If G is a (2t + 1)-edge-connected (2r + 1)-regular graph with
nvertices, then α′(G ) ≥ n2 − (
r−t2(r+1)2+t
)n2 , and this is sharp for t ≥ 1.
Proof. Let S be a set with maximum deficiency. Thus,α′(G ) = 12
(n − def(S)), where def(S) = o(G − S) − |S |.
Let ci count the odd components of G − S having i edges to S
.Let c = c(2t+1) + ... + c(2r−1), and let c
′ = o(G − S) − c .
Note that for (2t + 1) ≤ i ≤ 2r − 1, each odd component of G −
Shaving i edges to S has at least (2r + 3) vertices.(Otherwise,
each vertex of G has a neighbor outside.)
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Completion of Proof
Counting the edges joining S to odd components of G − S
yields(2r + 1)|S | ≥ (2r + 1)c ′ + (2t + 1)c .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Completion of Proof
Counting the edges joining S to odd components of G − S
yields(2r + 1)|S | ≥ (2r + 1)c ′ + (2t + 1)c .
Hence |S | ≥ c ′ + ( 2t+12r+1 )c ≥ (2t+12r+1 )c .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Completion of Proof
Counting the edges joining S to odd components of G − S
yields(2r + 1)|S | ≥ (2r + 1)c ′ + (2t + 1)c .
Hence |S | ≥ c ′ + ( 2t+12r+1 )c ≥ (2t+12r+1 )c .
Therefore, n ≥ |S | + c(2r + 3) ≥ ( 2t+12r+1 )c + c(2r + 3),
which implies that c ≤ ( 2r+14r2+4r+4+2t
)n. Now, we compute
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Completion of Proof
Counting the edges joining S to odd components of G − S
yields(2r + 1)|S | ≥ (2r + 1)c ′ + (2t + 1)c .
Hence |S | ≥ c ′ + ( 2t+12r+1 )c ≥ (2t+12r+1 )c .
Therefore, n ≥ |S | + c(2r + 3) ≥ ( 2t+12r+1 )c + c(2r + 3),
which implies that c ≤ ( 2r+14r2+4r+4+2t
)n. Now, we compute
def(S) = (c + c ′) − |S | ≤ c −2t + 1
2r + 1c =
2(r − t)
2r + 1c
≤2(r − t)
2r + 1
(
2r + 1
4r2 + 4r + 4 + 2t
)
n =(r − t)n
2(r + 1)2 + t.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Bounds for Other Cases
Theorem (O and West 2009+)
If G is a 2t-edge-connected (2r + 1)-regular graph with n
vertices,then again α′(G ) ≥ n2 − (
r−t2(r+1)2+t
)n2 , and this is sharp.
(same bound as when κ′(G ) ≥ 2t + 1)
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Bounds for Other Cases
Theorem (O and West 2009+)
If G is a 2t-edge-connected (2r + 1)-regular graph with n
vertices,then again α′(G ) ≥ n2 − (
r−t2(r+1)2+t
)n2 , and this is sharp.
(same bound as when κ′(G ) ≥ 2t + 1)
Theorem (O and West 2009+)
If G is a (2t − 1)-edge-connected 2r -regular graph with n
vertices,then α′(G ) ≥ n2 −(
r−t2r2+r+t
)n2 , and this inequality is sharp even whenG is
2t-edge-connected.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Sharpness for Odd Regular Graphs - Br ,t and Gr ,t
B2,1 a (5, 3)-biregular bigraph H G2,1
◮ Let Br ,t be a graph obtained from the graph Br by deleting
amatching with t edges in Br .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Sharpness for Odd Regular Graphs - Br ,t and Gr ,t
B2,1 a (5, 3)-biregular bigraph H G2,1
◮ Let Br ,t be a graph obtained from the graph Br by deleting
amatching with t edges in Br .
◮ To make a (2t + 1)-edge-connected (2r + 1)-regular Gr ,t
,replace each vertex of T in a (2t + 1)-edge-connected(2r +1, 2t
+1)-biregular (S ,T )-bigraph H with a copy of Br ,t .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Sharpness for Odd Regular Graphs - Br ,t and Gr ,t
B2,1 a (5, 3)-biregular bigraph H G2,1
◮ Let Br ,t be a graph obtained from the graph Br by deleting
amatching with t edges in Br .
◮ To make a (2t + 1)-edge-connected (2r + 1)-regular Gr ,t
,replace each vertex of T in a (2t + 1)-edge-connected(2r +1, 2t
+1)-biregular (S ,T )-bigraph H with a copy of Br ,t .
◮ Note: |S | = q(2t + 1) and |T | = q(2r + 1) for some q ∈
Q.
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Problem
Matching Number of Gr ,t
Proposition (O and West 2009+)
For 0 ≤ t ≤ r , α′(Gr ,t) =n2 − (
r−t2(r+1)2+t
)n2 ,
where n = q(2t + 1) + q(2r + 1)(2r + 3) = q(
(2r + 2)2 + 2t))
.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Matching Number of Gr ,t
Proposition (O and West 2009+)
For 0 ≤ t ≤ r , α′(Gr ,t) =n2 − (
r−t2(r+1)2+t
)n2 ,
where n = q(2t + 1) + q(2r + 1)(2r + 3) = q(
(2r + 2)2 + 2t))
.
Proof. By the previous theorem, α′(Gr ,t) ≥n2 − (
r−t2(r+1)2+t
)n2 .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Matching Number of Gr ,t
Proposition (O and West 2009+)
For 0 ≤ t ≤ r , α′(Gr ,t) =n2 − (
r−t2(r+1)2+t
)n2 ,
where n = q(2t + 1) + q(2r + 1)(2r + 3) = q(
(2r + 2)2 + 2t))
.
Proof. By the previous theorem, α′(Gr ,t) ≥n2 − (
r−t2(r+1)2+t
)n2 .
Recall that |S | = (2t + 1)q. Thus,
def(S) = o(G − S) − |S | = q(2r + 1) − q(2t + 1)
= 2q(r − t) = 2n
(2r + 2)2 + 2t(r − t) =
(r − t)n
2(r + 1)2 + t.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Edge-Connectivity of Br ,t
Lemma
For 0 ≤ t ≤ r , the edge-connectivity of the graph Br ,t is 2r
.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Edge-Connectivity of Br ,t
Lemma
For 0 ≤ t ≤ r , the edge-connectivity of the graph Br ,t is 2r
.
Proof.(Elementary exercise) If a graph G is connected, andn2 ≤
δ(G ) ≤ n, then κ
′(G ) = δ(G ), and the above is a special case.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Edge-Connectivity of Br ,t
Lemma
For 0 ≤ t ≤ r , the edge-connectivity of the graph Br ,t is 2r
.
Proof.(Elementary exercise) If a graph G is connected, andn2 ≤
δ(G ) ≤ n, then κ
′(G ) = δ(G ), and the above is a special case.
Lemma
Iteratively replacing a vertex of T with Br ,t in H preserves
(2t +1)-edge-connectedness.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Construction of Br ,t,k
To show that equality holds infinitely often, we need to build
aninfinite family of such H.
Cyclic construction using two copies of K3,5, and B2,1,2
We are gonna put a bunch of copies of K2t+1,2r+1 around a
circleand modify them to construct H.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Edge Connectivity of Br ,t,k
Lemma
For a ≥ b, if a graph H is the graph obtained from Ka,b by
deletinga matching of size b, then κ′(H) = b − 1.
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Edge Connectivity of Br ,t,k
Lemma
For a ≥ b, if a graph H is the graph obtained from Ka,b by
deletinga matching of size b, then κ′(H) = b − 1.
Proof.(Elementary exercise) If a graph G is a bipartite graph
withdiameter at most 3, then κ′(G ) = δ(G ), and the above is a
specialcase.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Edge Connectivity of Br ,t,k
Lemma
For a ≥ b, if a graph H is the graph obtained from Ka,b by
deletinga matching of size b, then κ′(H) = b − 1.
Proof.(Elementary exercise) If a graph G is a bipartite graph
withdiameter at most 3, then κ′(G ) = δ(G ), and the above is a
specialcase.
Theorem (O and West 2009+)
For 0 ≤ t ≤ r , κ′(Br ,t,k) = 2t + 1
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Sharpness for Even Regular Graphs : B ′r ,t and G′r ,t
Let B ′r ,t be the graph obtained from K2r+1 by deleting a
matchingof with t edges.To make a 2t-edge-connected 2r -regular
graph G ′r ,t , replace eachvertex of T in 2t-edge-connected (2r ,
2t)-biregular (S ,T )-bipartiteH ′ with a copy of B ′r
,t.Similarly, we can have an infinite family of 2t-edge-connected2r
-regular graphs like in previous steps.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem
A Chinese Postman tour in a connected graph G is a
shortestclosed walk traversing all edges in G .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem
A Chinese Postman tour in a connected graph G is a
shortestclosed walk traversing all edges in G .
Let eP(G ) be the number of edges in it.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem
A Chinese Postman tour in a connected graph G is a
shortestclosed walk traversing all edges in G .
Let eP(G ) be the number of edges in it.
A parity subgraph in a graph G is a spanning subgraph H of Gsuch
that dG (v) ≡ dH(v) (mod 2) for every vertex v in G .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem
A Chinese Postman tour in a connected graph G is a
shortestclosed walk traversing all edges in G .
Let eP(G ) be the number of edges in it.
A parity subgraph in a graph G is a spanning subgraph H of Gsuch
that dG (v) ≡ dH(v) (mod 2) for every vertex v in G .
Let p(G ), the parity number of G , be the minimum number
ofedges in a parity subgraph of G .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem
A Chinese Postman tour in a connected graph G is a
shortestclosed walk traversing all edges in G .
Let eP(G ) be the number of edges in it.
A parity subgraph in a graph G is a spanning subgraph H of Gsuch
that dG (v) ≡ dH(v) (mod 2) for every vertex v in G .
Let p(G ), the parity number of G , be the minimum number
ofedges in a parity subgraph of G .
Note that eP(G ) = |E (G )| + p(G ). In view of many
applicationsof the Chinese Postman problem, it is natural to ask
for the valueof eP(G ), or equivalently, the value of p(G ).
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Construction of H′r
a graph in T ′1 a graph in H′
1
◮ Let T ′r be the family of trees such that every non-leaf
vertexhas degree 2r + 1.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Construction of H′r
a graph in T ′1 a graph in H′
1
◮ Let T ′r be the family of trees such that every non-leaf
vertexhas degree 2r + 1.
◮ Let H′r be the family of (2r + 1)-regular graphs obtained
fromtrees in T ′r by identifying each leaf of such a tree with
theneck in a copy of Br .
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Construction of H′r
a graph in T ′1 a graph in H′
1
◮ Let T ′r be the family of trees such that every non-leaf
vertexhas degree 2r + 1.
◮ Let H′r be the family of (2r + 1)-regular graphs obtained
fromtrees in T ′r by identifying each leaf of such a tree with
theneck in a copy of Br .
◮ For r = 1, we will show the graph in H′r have the largest
valueof p(G ) among n-vertex cubic graphs.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Parity Number of H′r
Proposition
Let pr = 2r2 + 2r − 1. For any n-vertex graph G in H′r ,
b(G ) = (2r−1)n+22pr , c(G ) =r(n−2)−2
pr− 1.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Parity Number of H′r
Proposition
Let pr = 2r2 + 2r − 1. For any n-vertex graph G in H′r ,
b(G ) = (2r−1)n+22pr , c(G ) =r(n−2)−2
pr− 1.
Lemma
If G is regular of odd degree, then every cut-edge is in every
paritysubgraph.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Parity Number of H′r
Proposition
Let pr = 2r2 + 2r − 1. For any n-vertex graph G in H′r ,
b(G ) = (2r−1)n+22pr , c(G ) =r(n−2)−2
pr− 1.
Lemma
If G is regular of odd degree, then every cut-edge is in every
paritysubgraph.
Corollary
If G is a graph in H′r , and T is the tree obtained by shrinking
eachBr in G to one vertex, then every parity subgraph of G contains
T .
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Parity Number of H′r
Theorem (O and West 2009+)
If G is in H′r , then p(G ) =(2r2+3r−1)n−2(r+1)
4r2+4r−2− 1, which reduces to
2n−53 for cubic graphs.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Parity Number of H′r
Theorem (O and West 2009+)
If G is in H′r , then p(G ) =(2r2+3r−1)n−2(r+1)
4r2+4r−2− 1, which reduces to
2n−53 for cubic graphs.
Proof. Let T be the tree obtained by shrinking all the balloons
inG . By the previous lemma, a parity subgraph must use all
theedges in T .
O and West Generalized Balloons and Chinese Postman Problems
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Motivation and QuestionsPrevious Results using Balloons
Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Parity Number of H′r
Theorem (O and West 2009+)
If G is in H′r , then p(G ) =(2r2+3r−1)n−2(r+1)
4r2+4r−2− 1, which reduces to
2n−53 for cubic graphs.
Proof. Let T be the tree obtained by shrinking all the balloons
inG . By the previous lemma, a parity subgraph must use all
theedges in T . A parity subgraph of G must add r + 1 more edges
ineach balloon (since Br has 2r + 3 vertices).
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Parity Number of H′r
Theorem (O and West 2009+)
If G is in H′r , then p(G ) =(2r2+3r−1)n−2(r+1)
4r2+4r−2− 1, which reduces to
2n−53 for cubic graphs.
Proof. Let T be the tree obtained by shrinking all the balloons
inG . By the previous lemma, a parity subgraph must use all
theedges in T . A parity subgraph of G must add r + 1 more edges
ineach balloon (since Br has 2r + 3 vertices). Hence,
p(G ) = c(G ) + (r + 1)b(G ) = r(n−2)−2pr
− 1 + (r + 1) (2r−1)n+22pr
= 2r(n−2)−4+(r+1)(2r−1)n+22pr − 1 =(2r2+3r−1)n−2(r+1)
4r2+4r−2− 1.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Definitions and Remarks for the upper bound
◮ An r -graph is an r -regular multigraph G on an even number
ofvertices with the property that every edge-cut which separatesV
(G ) into two sets of odd cardinality has size at least r .
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Definitions and Remarks for the upper bound
◮ An r -graph is an r -regular multigraph G on an even number
ofvertices with the property that every edge-cut which separatesV
(G ) into two sets of odd cardinality has size at least r .
◮ Note that if G is a 2-edge-connected cubic multigraph, thenG
is 3-graph.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Definitions and Remarks for the upper bound
◮ An r -graph is an r -regular multigraph G on an even number
ofvertices with the property that every edge-cut which separatesV
(G ) into two sets of odd cardinality has size at least r .
◮ Note that if G is a 2-edge-connected cubic multigraph, thenG
is 3-graph.
◮ More generally, if G is an (r − 1)-edge-connected r
-regularmultigraph with even order, then G is r -graph.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Definitions and Remarks for the upper bound
◮ An r -graph is an r -regular multigraph G on an even number
ofvertices with the property that every edge-cut which separatesV
(G ) into two sets of odd cardinality has size at least r .
◮ Note that if G is a 2-edge-connected cubic multigraph, thenG
is 3-graph.
◮ More generally, if G is an (r − 1)-edge-connected r
-regularmultigraph with even order, then G is r -graph.
◮ Every r -edge-colorable r -regular graph is an r -graph.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Application of Edmonds’ Theorem
Lemma (Edmonds 1965)
If G is an r -graph, then there is an integer p and a family M
ofperfect matchings such that each edge of G is contained in
preciselyp members of M. (The members of M need not be
distinct.)
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Application of Edmonds’ Theorem
Lemma (Edmonds 1965)
If G is an r -graph, then there is an integer p and a family M
ofperfect matchings such that each edge of G is contained in
preciselyp members of M. (The members of M need not be
distinct.)
Lemma (O and West 2009+)
Let G be a 2r -edge-connected (2r + 1)-regular multigraph. If G
isedge-weighted, then there exists a perfect matching with weight
atmost 12r+1W , where W =
∑
e∈E(G) we and we is the weight on an
edge e. For cubic graphs, the bound reduces to 13W .
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Proof of the theorem
Proof. By Edmonds’ Lemma, |M|n2 =(2r+1)n
2 p, which implies that|M| = p(2r + 1).
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Proof of the theorem
Proof. By Edmonds’ Lemma, |M|n2 =(2r+1)n
2 p, which implies that|M| = p(2r + 1).
Let M = {M1, ...,Mp(2r+1)}.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Proof of the theorem
Proof. By Edmonds’ Lemma, |M|n2 =(2r+1)n
2 p, which implies that|M| = p(2r + 1).
Let M = {M1, ...,Mp(2r+1)}.
Since∑
wMi = p∑
e∈E(G) we = pW where wMi is the total weightof all edges in Mi ,
the pigeonhole principle implies that a matchingMj with the
smallest weight in the family has weight at most
12r+1W .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem in Cubic graphs
Fn: the family of connected cubic graphs with n vertices.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem in Cubic graphs
Fn: the family of connected cubic graphs with n vertices.
Theorem (O and West 2009+)
If G is in Fn and n ≥ 10, then p(G ) ≤2n−5
3 ,with equality if G ∈ H′1.
Proof. If G is in Fn and has no balloons or n = 10, then G has
aperfect matching and p(G ) = n/2 ≤ 2n−53 .
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem in Cubic graphs
Fn: the family of connected cubic graphs with n vertices.
Theorem (O and West 2009+)
If G is in Fn and n ≥ 10, then p(G ) ≤2n−5
3 ,with equality if G ∈ H′1.
Proof. If G is in Fn and has no balloons or n = 10, then G has
aperfect matching and p(G ) = n/2 ≤ 2n−53 .
Therefore, we may assume that G has a balloon and n > 10.
O and West Generalized Balloons and Chinese Postman Problems
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem in Cubic graphs
Fn: the family of connected cubic graphs with n vertices.
Theorem (O and West 2009+)
If G is in Fn and n ≥ 10, then p(G ) ≤2n−5
3 ,with equality if G ∈ H′1.
Proof. If G is in Fn and has no balloons or n = 10, then G has
aperfect matching and p(G ) = n/2 ≤ 2n−53 .
Therefore, we may assume that G has a balloon and n > 10.
Proceed by induction on n. Let e be a cut-edge.Let G1 and G2 be
the components of G − e.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
Problem
Chinese Postman Problem in Cubic graphs
Fn: the family of connected cubic graphs with n vertices.
Theorem (O and West 2009+)
If G is in Fn and n ≥ 10, then p(G ) ≤2n−5
3 ,with equality if G ∈ H′1.
Proof. If G is in Fn and has no balloons or n = 10, then G has
aperfect matching and p(G ) = n/2 ≤ 2n−53 .
Therefore, we may assume that G has a balloon and n > 10.
Proceed by induction on n. Let e be a cut-edge.Let G1 and G2 be
the components of G − e.
Let G ′1 and G′
2 be the graphs obtained from G by replacing G2 andG1,
respectively, with B1.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Chinese Postman Problem in Cubic Graphs
Every parity subgraph of G ′i contains e and and uses at least
twoedges in B1. Hence, p(G
′
i ) = p(Gi) + 3 andp(G ) = p(G ′1) + p(G
′
2) − 5.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Chinese Postman Problem in Cubic Graphs
Every parity subgraph of G ′i contains e and and uses at least
twoedges in B1. Hence, p(G
′
i ) = p(Gi) + 3 andp(G ) = p(G ′1) + p(G
′
2) − 5.
If neither G1 nor G2 is B1, then G′
1 and G′
2 are smaller than G .
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Chinese Postman Problem in Cubic Graphs
Every parity subgraph of G ′i contains e and and uses at least
twoedges in B1. Hence, p(G
′
i ) = p(Gi) + 3 andp(G ) = p(G ′1) + p(G
′
2) − 5.
If neither G1 nor G2 is B1, then G′
1 and G′
2 are smaller than G .
Letting ni = |V (G′
i )|, we have n = n1 + n2 − 10.By applying the induction
hypothesis to both G ′1 and G
′
2,
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Chinese Postman Problem in Cubic Graphs
Every parity subgraph of G ′i contains e and and uses at least
twoedges in B1. Hence, p(G
′
i ) = p(Gi) + 3 andp(G ) = p(G ′1) + p(G
′
2) − 5.
If neither G1 nor G2 is B1, then G′
1 and G′
2 are smaller than G .
Letting ni = |V (G′
i )|, we have n = n1 + n2 − 10.By applying the induction
hypothesis to both G ′1 and G
′
2,
p(G ) = p(G ′1) + p(G′
2) − 5 ≤2n1 − 5
3+
2n2 − 5
3− 5 =
2n − 5
3
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Chinese Postman Problem in Cubic Graphs
1 1 1 3
Last case : every cut-edge is incident to a copy of B1.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Chinese Postman Problem in Cubic Graphs
1 1 1 3
Last case : every cut-edge is incident to a copy of B1.Let each
edge have weight 1. Form G ′ by deleting all the balloons.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Chinese Postman Problem in Cubic Graphs
1 1 1 3
Last case : every cut-edge is incident to a copy of B1.Let each
edge have weight 1. Form G ′ by deleting all the balloons.In G ′,
replace each ′′thread ′′ through vertices of degree 2 with asingle
edge whose weight is the length of the thread.
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Chinese Postman Problem in Cubic Graphs
1 1 1 3
Last case : every cut-edge is incident to a copy of B1.Let each
edge have weight 1. Form G ′ by deleting all the balloons.In G ′,
replace each ′′thread ′′ through vertices of degree 2 with asingle
edge whose weight is the length of the thread.The resulting
weighted graph G ′′ has a perfect matching with atmost 1/3 of its
total weight(=m−8b3 ).(Special case : when G ′ is a cycle, G has a
perfect matching.)
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Smallest Matching Number and Edge-ConnectivityChinese Postman
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Chinese Postman Problem in Cubic Graphs
1 1 1 3
Last case : every cut-edge is incident to a copy of B1.Let each
edge have weight 1. Form G ′ by deleting all the balloons.In G ′,
replace each ′′thread ′′ through vertices of degree 2 with asingle
edge whose weight is the length of the thread.The resulting
weighted graph G ′′ has a perfect matching with atmost 1/3 of its
total weight(=m−8b3 ).(Special case : when G ′ is a cycle, G has a
perfect matching.)Now, p(G ) ≤ p(G ′) + 3b ≤ m−8b3 + 3b =
3n−16b6 + 3b ≤
2n−53 .
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Matching Number and Edge-ConnectivityChinese Postman Problem