On the palette index of complete bipartite graphs

Abstract

The palette of a vertex x of a graph G determined by a proper edge colouring ϕ of G is the set {ϕ(xy) : xy ∈ E(G)} and the diversity of ϕ is the number of different palettes determined by ϕ. The palette index of G is the minimum of diversities of ϕ taken over all proper edge colourings ϕ of G. In the article we determine the palette index of K m,n for m ≤ 5 and pose two conjectures concerning the palette index of complete bipartite graphs.

A family of multigraphs with large palette index

“…• [7] Let K m,n be a complete bipartite graph with 1 ≤ m ≤ n. This situation is a little more involved, in the sense that we cannot always obtain a good upper bound for χ š (K m,n ) using the proofs of the results in [7]. In some cases, see for instance Proposition 11 in [7], the number of colors is twice the maximum degree ∆ (recall that minimizing the number of colors was not important in that context). Nevertheless, we analyze some small cases and obtain the same number of palettes (the minimum) by using a smaller number of colors.…”

“…One such example is obtained by considering the graph K 5,6 (i.e. case k = 3 in Proposition 11 of [7]). Denote by {u 1 , .…”

“…, v 6 } the bipartition of the vertex-set of K 5,6 . The proof of Proposition 11 in [7] furnishes an edge-coloring with 12 colors and 6 palettes. Following the notation used in [7] we represent the coloring with a matrix, where the element in position (i, j) is the color of the edge u i v j .…”

“…• A connected class 2 cubic graph has palette index 3 or 4 according as it does or it does not possess a perfect matching, respectively [8,Theorem 9]. • The palette index of complete bipartite graphs was determined in [7] in many instances.…”

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.…”