If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote the number
of vertices of $G$ and $Delta=Delta(G)$ the maximum degree of a vertex in $G$. We prove that a graph $G$ of order at least 6 is equitably $Delta$-colorable if $G$ satisfies $(|G|+1)/3 leq Delta < |G|/2$ and none of its components is a $K_{Delta +1}$.
Let $m$, $n$, and $k$ be integers satisfying $0 < k leq n < 2k leq m$. A family of sets $mathcal{F}$ is called an $(m,n,k)$-intersecting family if $binom{[n]}{k} subseteq mathcal{F} subseteq binom{[m]}{k}$ and any pair of members of $mathcal{F}$ have
nonempty intersection. Maximum $(m,k,k)$- and $(m,k+1,k)$-intersecting families are determined by the theorems of ErdH{o}s-Ko-Rado and Hilton-Milner, respectively. We determine the maximum families for the cases $n = 2k-1, 2k-2, 2k-3$, and $m$ sufficiently large.
We confirm the equitable $Delta$-coloring conjecture for interval graphs and establish the monotonicity of equitable colorability for them. We further obtain results on equitable colorability about square (or Cartesian) and cross (or direct) products of graphs.
