Graphs – Basic Concepts William T. Trotter and Mitchel T. Keller Math 3012 Applied Combinatorics Spring 2009.

Graphs – Basic Concepts

William T. Trotter and Mitchel T. KellerMath 3012 Applied Combinatorics

Spring 2009

What is a Graph?

What is a Graph?

A graph G is a pair (V, E) where V is a set (almost always finite in this course) and E is a collection of 2-element subsets of V.

Elements of V are called vertices and V is the vertex set.

Elements of E are called edges and E is the edge set

Notation and Terminology

Usually, we write xy is an edge in G, or xy E rather than {x,y} E.

Of course, xy is an edge if and only if yx is an edge. When xy is an edge, we say x and y are adjacent. When x and y are distinct and xy is not an edge, we

say that x and y are non-adjacent.

Data Files for Graphs

graph_data.txt63 24 65 66 17 36 2

Subgraphs – Two kinds

H is a subgraph of G when every vertex of H is a vertex in G, and every edge in H is an edge in G.

NO YES

Induced Subgraphs

H is an induced subgraph of G when every vertex of H is a vertex in G, and every edge in G with both endpoints in H is an edge in H.

NO YES

Isomorphic Graphs

G and H are isomorphic when there is a bijection f : V(G) V(H) between their vertex sets so that x and y are adjacent in G if and only if f(x) and f(y) are adjacent in H.

Paths in Graphs

A path in a graph (from x to y) is a sequence x0, x1, x2, …, xt such that

1. x = x0;

2. y = xt; and

3. xi xi+1 is an edge for every i = 0,1,2,…, t -1.

A Path from 18 to 12

Connected Graphs

A graph G is connected if there is a path from x to y for every distinct pair of vertices in G.

A Connected Graph

Components of Disconnected Graphs

When a graph is disconnected (not connected), an induced subgraph H is called a component of G when:

(1) H is connected; and

(2) there is no connected subgraph of G containing the vertex set of H and having more vertices than H.

A Disconnected Graph with 3 components

Cycles in Graphs

A cycle in a graph G (from x to y) is a sequence x0, x1, x2, …, xt of distinct vertices from G with t ≥ 2 such that

1. x = x0;

2. y = xt; and

3. xi xi+1 is an edge for every i = 0,1,…, t -1;

4. xt x0 is an edge.

Lenth of Paths and Cycles

When (x0, x1, x2, …, xt) is a path, there are t+1 vertices in the sequence, but we say that the length of the path is t. This counts the number of edges.

For a positive integer n, it is customary to denote a path on n vertices as Pn. The length of Pn is then n-1.

When (x0, x1, x2, …, xt) is a cycle, there are t+1 vertices in the sequence, but now we say that the length of the cycle is t+1. This again counts the number of edges since the last vertex is also adjacent to the first.

For a positive integer n, it is customary to denote a cycle on n vertices as Cn. The length of Cn is then n.

A cycle of length 8

Complete and Independent Graphs

Cliques in Graphs

A set S of vertices in a graph G is called a clique when every distinct pair of vertices in S is adjacent.

A clique in G is just a set of vertices that induces a complete subgraph of G.

The maximum clique size of G is denoted by ω(G).

Triangles in Graphs

A clique of size 3 is called a triangle.

{1,2,8} is a triangle, but ω(G) = 4

ω(G) = 6

Caution: Are we certain that ω(G) = 6

To show that ω(G) ≥ 6, it is enough to show that G contains a clique of size 6.

But how do we show that G does not contain a clique of size 7 without testing every subset S consisting of 7 vertices of G? If G contains 34824125 vertices, this could take a long time!

Determining ω(G)

Alice claims that ω(G) = 257. How does she verify this claim for a graph G with 10342 vertices?

Can you write C code that will accept a graph data file as input and output a text file, which has ω(G) as an integer on the first line followed by ω(G) integers, one per line, immediately below? If you can do this and have your algorithm run in time which is polynomial in the input size, then you are guaranteed an A++ in this course!! Please share your code with Professor Trotter before going public.

Independent Sets in Graphs

A set S of vertices in a graph G is called an independent set (also a stable set) when no distinct pair of vertices in S is adjacent in G.

The maximum size of an independent set of vertices in G is called the independence number of G and is

denoted α(G).

α(G) = 12

Are you certain that α(G) = 12

Same caution as before. We have shown only that α(G) ≥ 12. It requires much more work to show that G does not contain an independent set of size 13.

Graph Coloring

A t - coloring of a graph is a function f which assigns to each vertex x an integer f(x) from {1,2,…,t} so that f(x) ≠ f(y) whenever xy is an edge of G.

The chromatic number of G, denoted Χ(G), is the least positive integer t for which G has a t – coloring.

A 6 - Coloring

This shows that Χ(G) ≤ 6

Χ(G) ≤ 4

A Trivial Inequality

Χ(G) ≥ ω(G)

The chromatic number of a graph is at least as large as the maximum clique size.

Χ(G) = ω(G) = 4

How Good is this Inequality?

Χ(G) ≥ ω(G)

Maybe, just maybe, the chromatic number of a graph is always equal to the maximum clique size?!

Not Always Equal

3 = Χ(G) > ω(G) = 2

Very large Χ and small ω

Theorem. For every t ≥ 3, there exists a graph Gt so that

Χ(G) = t and ω(G) = 2

Posets and Cover Graphs

Cover graphs of posets are triangle-free, i.e., ω(G) ≤ 2

Height and Chromatic Number

If G is the cover graph of a poset P with

height (P) = h,

then:

Χ(G) ≤ h

Since any partition of P into h antichains is also a coloring of G using h colors.

Posets and large Χ, small ω

Theorem. For every t ≥ 3, there exists a poset Pt with height (Pt) = t so that if Gt is the cover graph of Pt, then

Χ(Gt) = t and ω(Gt) = 2

The First Case

3 = height(P3) = Χ(G3) while ω(G3) = 2

P3