Incidence Coloring in Graph Theory: Coloring Vertices and Edges

Explore incidence coloring in graph theory, a variation of graph coloring where colors are assigned to incidences (vertex-edge pairs). This guide defines incidence coloring, explains the incidence chromatic number, and discusses different types of incidence colorings.

Incidence Coloring in Discrete Mathematics

Introduction to Incidence Coloring

Incidence coloring is a concept in graph theory where we color the "incidences" of a graph. An incidence is a pair (v, e) where 'v' is a vertex (a point) and 'e' is an edge (a line connecting two vertices) such that the edge 'e' is connected to the vertex 'v'. The goal is to assign colors to these incidences so that no two adjacent incidences have the same color. Two incidences are adjacent if they share a vertex, share an edge, or if their vertex and edge combinations overlap.

Incidence Chromatic Number (χi(G))

The incidence chromatic number χi(G) of a graph G is the smallest number of colors needed for a proper incidence coloring.

Example: Incidence Coloring of C₄

(An illustrative figure showing the cycle graph C₄ and its incidence coloring should be included here. The figure should show that at least 4 colors are needed for a proper coloring.)

Important Notations

  • V(G): The set of vertices in graph G.
  • E(G): The set of edges in graph G.
  • I(G): The set of incidences in graph G (pairs (v, e) where vertex v is incident with edge e).
  • deg(v): The degree of vertex v (number of edges connected to v).
  • χi(G): The incidence chromatic number of G.
  • A+(v): Incidences of the form (u, uv) for vertex v.
  • A-(v): Incidences of the form (v, vu) for vertex v.

(k, p)-Incidence Coloring

A (k, p)-incidence coloring uses at most k colors, with at most p colors used for incidences in A+(v) for each vertex v. χp(G) is the smallest number of colors needed for a (k, p)-incidence coloring. Note that χi(G) ≤ χp(G) for all p ≥ 1.

Connections to Other Graph Concepts

  • Incidence Graph: The incidence graph IG(G) has vertices representing the incidences of G, and edges connecting adjacent incidences. An incidence coloring of G is a proper vertex coloring of IG(G).
  • Bipartite Graph: The bipartite graph H, with V(H) = V(G) ∪ E(G) and edges connecting vertices and incident edges, relates to incidence coloring through strong edge coloring.
  • Arc Coloring: In an undirected graph, creating a directed graph G* by replacing each edge with two opposite arcs, incidence coloring of G corresponds to arc coloring of G* under certain conditions. (These conditions are described in the original text and would be included here for completeness.)

(k, 1)-Incidence Coloring

In a (k, 1)-incidence coloring, for each vertex v, only one color is used for incidences in A+(v). This implies that for any two vertices u and v with distance 1 or 2, they must have different colors. The mapping c(v) = f(u, uv) is well-defined under these conditions.

Conclusion

Incidence coloring is a fascinating area of graph theory. While seemingly simple, it has deep connections to other graph coloring problems and offers unique challenges in finding optimal colorings.