On multiples of simple graphs and Vizing's Theorem

J.M. McDonald
2010 Discrete Mathematics  
Let G be a simple connected graph with maximum degree d, and let tG denote the graph obtained from G by replacing each edge with t parallel edges. Vizing's Theorem says that td ≤ χ (tG) ≤ td + t. When t = 1, i.e., when tG is a simple graph, Holyer proved that it is NP-hard to decide if χ (tG) = td + 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 This characterization is best possible in the sense that for any
more » ... r of positive integers t and d with t ≤ d/2, there exist non-complete simple connected graphs G with maximum degree d and χ (tG) = td + t. A multiple of a simple graph G is a multigraph obtained from G by replacing every edge with the same number of parallel edges. If each edge is replaced by t parallel edges, then we call the resulting multigraph tG. Note that if G has maximum degree d, then tG has maximum degree td. So, the chromatic index χ of tG (the minimum number of colours needed to colour the edges of tG such that adjacent edges receive different colours), is at least td. Vizing's Theorem [9] tells us that in fact, td ≤ χ (tG) ≤ td + t. When t = 1, i.e., when tG is a simple graph, Holyer [2] proved that it is NP-hard to decide if χ (tG) = td + t or not. Here we show that when t > d/2 it is not NP-hard to answer this question, and in fact we provide the following simple
doi:10.1016/j.disc.2010.04.012 fatcat:gdpx4w2iqrg4vfuf6wghwzlod4