On multiples of simple graphs and Vizing’s Theorem

Abstract

Let G be a simple connected graph with maximum degree d , and let t G denote the graph obtained from G by replacing each edge with t parallel edges. Vizing’s Theorem says that t d ≤ χ ′ ( t G ) ≤ t d + t . When t = 1 , i.e., when t G is a simple graph, Holyer proved that it is NP-hard to decide if χ ′ ( t G ) = t d + t or not. Here we show, using a recent result of Scheide, that when t > d / 2 it is not NP-hard to answer this question, and in fact χ ′ ( t G ) = t d + t if and only if G = K d + 1 and d is even. This characterization is best possible in the sense that for any pair of positive integers t and d with t ≤ d / 2 , there exist non-complete simple connected graphs G with maximum degree d and χ ′ ( t G ) = t d + t .