In this note, we derive an explicit formula for the equitable chromatic number of a complete n-partite graph Kp1 ;p2 ;:::;pn . Namely, if M is the largest integer such that pi (modM)¡ ⌈pi M ⌉ (i = 1; 2; : : : ; n) then e(Kp1 ;p2 ;:::;pn) = n ∑

We propose the following conjecture to generalize results of Pósa and Corrádi Hajnal. Let r, s be nonnegative integers and let G be a graph with |V (G)| ≥ 3r + 4s and minimal degree δ(G) ≥ 2r + 3s. Then G contains a collection of r + s vertex disjoint cycles, s of them with a chord. We prove the conjecture for r = 0, s = 2 and for s = 1. The corresponding extremal problem, to find the minimum n...

A graph G is r-equitably k-colorable if its vertex set can be partitioned into k independent sets, any two of which differ in size by at most r. The r-equitable chromatic threshold of a graph G, denoted by χr=(G), is the minimum k such that G is r-equitably k -colorable for all k ≥ k. Let G × H denote the Kronecker product of graphs G and H. In this paper, we completely determine the exact valu...