The Vertex Arboricity of The Vertex Arboricity of Integer Distance Graph Integer Distance Graph with a Special Distance with a Special Distance Set Set Juan Liu* and Qinglin Yu Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China
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The Vertex Arboricity of Integer Distance Graph with a Special Distance Set
The Vertex Arboricity of Integer Distance Graph with a Special Distance Set. Juan Liu * and Qinglin Yu Center for Combinatorics, LPMC Nankai University, Tianjin 300071, P. R. China. Outline. Definitions and Notations Background and K nown R esults Main Theorem. Definitions and Notations. - PowerPoint PPT Presentation
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The Vertex Arboricity of Integer The Vertex Arboricity of Integer Distance Graph with a Special Distance Graph with a Special
Distance SetDistance Set
Juan Liu* and Qinglin Yu
Center for Combinatorics, LPMCNankai University, Tianjin 300071, P. R. China
OutlineOutline
Definitions and NotationsBackground and Known ResultsMain Theorem
Definitions and NotationsDefinitions and Notations
Vertex arboricity
Given a graph G, a k-coloring of G is a mapping from V(G) to [1, k].
denotes the set of all vertices of G colored with i, and denotes the subgraph induced by in G.
iV
iV
iV
Chromatic Number VS Vertex Arboricity Chromatic Number VS Vertex Arboricity
a proper k-coloring:
each is an
independent set. chromatic number
= min {k|G has a
proper k-coloring}
a tree k-coloring:
each induces a forest.
vertex arboricity va(G)
va(G) = min {k|G has a
tree k-coloring }
( )G
( )G
iV iV
Vertex ArboricityVertex Arboricity
Vertex arboricity is the minimum number of subsets into which V(G) can be partitioned so that each subset induces an acyclic subgraph of G.
Clearly,
for any graph G.( ) ( )va G G
ExamplesExamples
( ) 2va G =
( ) 5Gc =
5( ) 3va K =
(G) 3c =
( ) 3va G =
5(K ) 5c =
Known results for Known results for vava(G)(G)
(Kronk & Mitchem, 1975) For any graph G,
(Catlin & Lai, 1995) If G is neither a cycle nor a clique, then
( ) 1( )
2
Gva G
( )( )
2
Gva G
Known results for Known results for vava((GG))
(Skrekovski, 1975) For a locally planar
graph G, ; For a triangle-free
locally planar graph G, .
(Jorgensen, 2001) Every graph without a
-minor has vertex arboricity at most 4.
( ) 3va G ( ) 2va G
4,4K
Definitions and NotationsDefinitions and Notations
Distance graph If and , then the distance graph G(S, D) is defined by the graph with vertex set S and two vertices x and y are adjacent if and only if
where the set D is called the distance set.
S D
| |x y D
Definitions and NotationsDefinitions and Notations
Integer distance graph
if and all elements of D are positive integers, then the graph G(Z, D)=G(D) is called the integer distance graph and the set D is called its integer distance set.
S
Examples ofExamples of Integer Distance GraphInteger Distance Graph
D={2} -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
D={1, 3}
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
BackgroundBackground
The distance graph was introduced by Eggleton et al in 1985.
Coloring problems on distance graphs are motivated by the famous Hadwiger-Nelson coloring problem on the unit distance plane.
Known resultsKnown results
Chromatic number of integer distance graph;
Vertex arboricity of integer distance graph.
Results onResults on
(Eggleton, Erdos & Skilton, 1984)
where D is an interval
between 1 and for .
( ( , )) 2G R D n
1 1n n
( , )G R D
Results onResults on
(Eggleton, Erdos & Skilton, 1985) If a and b are relatively prime positive integers of opposite parity, then .