Abstract
A proper edge-coloring of a graph G is an assignment of colors to the edges of G such that adjacent edges receive distinct colors. A proper edge-coloring defines at each vertex the set of colors of its incident edges. Following the terminology introduced by Hor?ak, Kalinowski, Meszka and Wo?niak, we call such a set of colors the palette of the vertex. What is the minimum number of distinct palettes taken over all proper edge-colorings of G? A complete answer is known for complete graphs and cubic graphs. We study in some detail the problem for 4-regular graphs. In particular, we show that certain values of the palette index imply the existence of an even cycle decomposition of size 3 (a partition of the edge-set of a graph into 3 2-regular subgraphs whose connected components are cycles of even length). This result can be extended to 4d-regular graphs. Moreover, in studying the palette index of a 4-regular graph, the following problem arises: does there exist a 4-regular graph whose even cycle decompositions cannot have size smaller than 4?
Highlights
Throughout this paper, a graph G always means a simple finite graph
An edge-coloring of a graph G is an assignment of colors to the edges of G: it is proper if adjacent edges receive distinct colors
A natural question about even cycle decompositions of 4-regular graphs arise from these considerations: Does there exist a 4-regular graph with all even cycle decompositions of size larger than 3? In Section 2.3 we study connected 4-regular graphs with palette index 4
Summary
Throughout this paper, a graph G always means a simple finite graph (without loops and parallel edges). From an even cycle decomposition of size 3 of a 4-regular graph G we can obtain a proper edge-coloring of G with exactly 3 palettes (see Proposition 11).
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