Some bounds on the number of colors in interval and cyclic interval edge colorings of graphs

Abstract

An interval t-coloring of a multigraph G is a proper edge coloring with colors 1, . . . , t such that the colors on the edges incident to 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 on the edges incident to 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) (w c (G)) and W (G) (W c (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 (∆(G) − 1). We also give several results towards the general conjecture that W c (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

Interval colorings of graphs—Coordinated and unstable no‐wait schedules

“…However, they are present on other two edges incident to v′. The respective vertices in V 0 are incident to two edges labeled by consecutive integers: D D (8, 7), (9,8), (10,9), (11,12), (12,11), …, ( + 19, + 20).…”

“…Indeed, otherwise considering the labels in an interval coloring modulo $${\rm{\Delta }}(G)$$ gives a proper edge‐coloring using at most $${\rm{\Delta }}(G)$$ colors. Interval colorings for special classes of graphs and related problems were considered, see, for example, [2–4, 6, 7, 9–13, 16–21, 24, 25].…”

“…11,12, 14 + , 16 + , 18 + , 25 + , respectively. Since the label of the last edge is exactly D+ 25, we see that all the edges of P′ have exactly the labels listed:11, 12, 14 + , 16 + , 18 + , 25 + .…”

A Characterization of Graphs with Small Palette Index

“…Further results on the palette index of a graph appeared since then (see, e.g., [2][3][4][5][6][7][8]). Many of them consider the computation of the palette index for some specific classes of graphs, such as complete bipartite graphs, 4-regular graphs and some others.…”

Some results on the palette index of graphs

“…Note further that it is in fact an open problem to determine if there is a graph G that requires more than ∆(G) + 1 colors for a cyclic interval coloring (cf. Casselgren et al (2018)).…”