The total coloring of F<inf>m</inf> ∨ F<inf>n</inf>

Abstract

Results on graph coloring can be used to draw conclusions about scheduling. Graph theory is a sort of models which can be applied in various science fields such as computer science, physics, biology, chemistry, strategy etc. And graph coloring is one of the chief topics in graph research. Suppose G(V,E) is a connect graph with order at least 2, k is a positive integer and ƒ is the mapping from V (G)∪E(G) to {1, 2, ⋯, k}. If (1) for any uv, vw ∈ E(G), u ≠ w, we have ƒ(uv) ≠ ƒ(vw); (2) for any uv ∈ E(G), u ≠ v, we have ƒ(u) ≠ ƒ(v), ƒ(u) ≠ ƒ(uv), ƒ(v) ≠ f(uv), then ƒ is called a k-total coloring of graph G(denoted by k - TC of G). The total chromatic number, denoted by χt (G), is the least number of colors in a total coloring of graph G. Suppose G and H are two simple graphs, (V (G)∪E(G))∩(V (H)∪E(H)) = ∅. Let V (G∨H) = V (G)∪V (H), E(G∨H) = E(G)∪E(H)∪{uv∣u ∈ V (G), v ∈ V (H)}, then G∨H is called the join-graph of G and H. The total chromatic number of the join graph of two fans with orders m + 1 and n + 1 respectively is obtained in this paper.