Minimum Number of Palettes in Edge Colorings

Abstract

A proper edge-coloring of a graph defines at each vertex the set of colors of its incident edges. This set is called the palette of the vertex. In this paper we are interested in the minimum number of palettes taken over all possible proper colorings of a graph.

Some results on the palette index of graphs

“…Trivially, š(G) = 1 if and only G is a regular Class 1 graph, and by Vizing's edge coloring theorem it holds that if G is regular and Class 2, then 3 ≤ š(G) ≤ ∆(G) + 1; the case š(G) = 2 is not possible, as proved in Horňák et al (2014).…”

“…Given an edge coloring ϕ of a graph G, we define the palette S G (v, ϕ) (or just S(v, ϕ)) of a vertex v ∈ V (G) as the set of all colors appearing on edges incident with v. The palette index š(G) of G is the minimum number of distinct palettes occurring in a proper edge coloring of G. This notion was introduced quite recently by Horňák et al (2014) and has so far primarily been studied for the case of regular graphs.…”

“…Note further that this in fact means that even determining if a given graph has palette index 1 is an N P-complete problem. Nevertheless, in Horňák et al (2014) it was proved that the palette index of a cubic Class 2 graph is 3 or 4 according to whether the graph has a perfect matching or not. Bonvicini and Mazzuoccolo (2016) investigated 4-regular graphs; they proved that š(G) ∈ {3, 4, 5} if G is 4-regular and Class 2, and that all these values are in fact attained.…”

Graphs with large palette index

“…The palette index š(G) of a graph G is the minimum number of distinct palettes, taken over all edge-colorings, occurring among the vertices of the graph. This parameter was formally introduced in [8] and several results have appeared since then, see [2,4,5,6,7,9]. All mentioned papers mainly consider the computation of the palette index in some special classes of graphs, such as trees, complete graphs, complete bipartite graphs, 3− and 4−regular graphs and some others.…”

“…It is an easy consequence of the definition that a graph has palette index equal to 1 if and only if it is a k-regular graph admitting a k-edge-coloring. Moreover, it is proved in [8] that no regular graph has palette index equal to 2. The situation is less understood when we ask for r-regular graphs with a large palette index.…”