A QUESTION ON RELAXED EQUITABLE COLORING

Abstract

An equitable (k, d)-coloring is a (k, d)-coloring that is also equitable. In this paper, we raise a conjecture that if G is a graph such that for each edge xy ∈ E(G), the sum d(x) + d(y) at most 2r + 1, then G has a d-relaxed equitable coloring with r + 1 - d colors. We prove that if graph G meet d(x) + d(y) ≤ 2r for every edge xy ∈ E(G) and |G| = rs, then [Formula: see text].