Graph Theory Lecture Notes 10

Vertex Colorings

Definitions

A vertex coloring of a graph is an assignment of colors (actually any set of labels), one to each vertex of the graph.

A proper vertex coloring is a vertex coloring with the property that adjacent vertices get different colors.

A graph is (vertex) k-colorable if it has a proper coloring using exactly k colors.

The (vertex) chromatic number of a graph G, denoted (G), is the minimum number of different colors required for a proper (vertex) coloring of G.

Thus, (G) = k if graph G is k-colorable but not (k-1)-colorable.

Applications

  1. Assignment of Radio Frequencies

  2. Scheduling Problems

Calculating Chromatic Numbers

Prop 10.1.2: Let G be a graph with k mutually adjacent vertices. Then (G)k.

The clique number(G) of a graph G is the number of vertices in a largest clique in G.

Corollary 10.1.3: Let G be a graph. Then (G)(G).

Prop 10.1.4: Let H be a subgraph of G. Then (G)(H).

Chromatic Numbers of Common Graph Families

Graph G(G)
complete graph Knn
bipartite graph2
non-trivial path graph Pn2
non-trivial tree2
hypercube graph Qn 2
cyclic graph Cn, n even2
cyclic graph Cn, n odd3
wheel graph Wn, n odd3
wheel graph Wn, n even4

Vertex Coloring Algorithms

Deciding whether or not a graph has a 3-coloring is an NP-complete problem, so in particular there is no known algorithm for coloring a graph with the minimum number of colors which runs in polynomial time. We will look at two algorithms which run quickly to produce a proper vertex coloring, but give no guarantee that the coloring is minimal.
  1. Sequential vertex coloring
  2. Largest Degree First

Brooks's Theorem

We denote the largest degree of any vertex in a graph G by max(G).

Theorem 10.1.18: (G)max(G)+ 1.

Proof: This follows from the sequential vertex coloring algorithm.

Theorem 10.1.20: (Brooks, 1941) Let G be a non-complete, simple connected graph with max(G)3. Then (G)max(G).

Map Coloring

Given a map with various regions in it, we wish to color the regions so that no two regions with a common border (consisting of more than one point) are given different colors. This map coloring problem can be restated in terms of graphs by using Poincaré duality. That is, each region of the map is a vertex of a graph and two vertices are joined by an edge if and only if the regions have a common border. Coloring the map is then equivalent to finding a proper vertex coloring of this graph. The graphs that arise this way are planar graphs, graphs which can be drawn with no edges meeting except at vertices.

Theorem 10.2.2: The Five-Color Theorem (Heawood, 1890). The chromatic number of a planar simple graph is at most 5.

Theorem 10.2.3: The Four-Color Theorem (Appel and Haken, 1976). Every planar graph is 4-colorable. We will look at the BBC CD called Only four colours.