For loopless multigraphs, the total chromatic number is asymptotically its fractional counterpart as the latter invariant tends to infinity. The proof of this is based on a recent theorem of Kahn establishing the analogous asymptotic behaviour of the list-chromatic index for multigraphs. The total colouring conjecture, proposed independently by Behzad [1] and Vizing [11], asserts that the total...

Guantao Chen Department of Mathematics and Computing Science Georgia State University ATLANTA GA 30303 USA Given a graph theoretic parameter f, a graph H and a positive integer m, the mixed ramsey number f(m,H) is defmed as the least positive integer p such that for any graph G of order p either f(G) ~ m or G contains H as a subgraph. In this paper we determine the mixed ramsey number X2(m,K( I...

The minimum number of total independent sets of V ∪ E of graph G(V,E) is called the total chromatic number of G, denoted by χ′′(G). If difference of cardinalities of any two total independent sets is at most one, then the minimum number of total independent partition sets of V ∪E is called the equitable total chromatic number, and denoted by χ′′ =(G). In this paper we consider equitable total c...

If G is a simple graph with minimum degree <5(G) satisfying <5(G) ^ f(| K(C?)| -f-1) the total chromatic number conjecture holds; moreover if S(G) ^ f| V(G)\ then #T(G) < A(G) + 3. Also if G has odd order and is regular with d{G) ^ \^/1\V{G)\ then a necessary and sufficient condition for ^T((7) = A((7)+ 1 is given.

The total chromatic number z t (G) of a graph G is the least number of colors needed to color the vertices and edges of G so that no adjacent vertices or edges receive the same color, no incident edges receive the same color as either of the vertices it is incident with. In this paper, we obtain some results of the total chromatic number of the equibipartite graphs of order 2n with maximum degr...

The total chromatic number, Z"(G), of a graph G, is defined to be the minimum number ofcolours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that )('(G) _< z'(G) + 2x/~G), where z(G)is the vertex chromatic number and )((G)is the edge chromatic number of the...

Let G = (V (G), E(G)) be a simple graph and T (G) be the set of vertices and edges of G. Let C be a k−color set. A (proper) total k−coloring f of G is a function f : T (G) −→ C such that no adjacent or incident elements of T (G) receive the same color. For any u ∈ V (G), denote C(u) = {f(u)} ∪ {f(uv)|uv ∈ E(G)}. The total k−coloring f of G is called the adjacent vertex-distinguishing if C(u) 6=...