Casselgren Carl Johan
Khachatrian Hrant
Petrosyan Petros
An interval t-coloring of a multigraph G is a proper edge coloring with colors 1,…,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors. A cyclic interval t-coloring of a multigraph G is a proper edge coloring with colors 1,…,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. Denote by w(G) (wc(G)) and W(G) (Wc(G)) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph G, respectively. We present some new sharp bounds on w(G) and W(G) for multigraphs G satisfying various conditions. In particular, we show that if G is a 2-connected multigraph with an interval coloring, then W(G)≤1+[V(G)/2](Δ(G)−1). We also give several results towards the general conjecture that Wc(G)≤|V(G)| for any triangle-free graph G with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most 4.
Springer
English
Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs
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