Some results on cyclic interval edge colorings of graphs

Abstract

A proper edge coloring of a graph G with colors 1, 2, . . . , t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree ∆(G) ≥ 4 admits a cyclic intervalWe also prove that every Eulerian bipartite graph G with maximum degree at most 8 has a cyclic interval coloring. Some results are obtained for (a, b)-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)-biregular graphs as well as all (2r −2, 2r)-biregular (r ≥ 2) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.

Cyclic deficiency of graphs

“…Generally, the problem of determining whether a given bipartite graph admits a cyclic interval coloring is NP -complete [28]. Nevertheless, all graphs with maximum degree at most 3 [29], complete multipartite graphs [2], outerplanar bipartite graphs [36], bipartite graphs with maximum degree 4 and Eulerian bipartite graphs with maximum degree not exceeding 8 [2], and some families of biregular bipartite graphs [10,11,2] admit cyclic interval colorings.…”

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

“…[18,19,22,1]. In particular, the general question of determining whether a graph G has a cyclic interval coloring is N P-complete [18] and there are concrete examples of connected graphs having no cyclic interval coloring [19,22,1]. Trivially, any multigraph with an interval coloring also has a cyclic interval coloring with ∆(G) colors, but the converse does not hold [19].…”

“…For the case d = 1, this reformulated problem has a positive answer for ordinary graphs (see e.g. [22,1]).…”

One-sided interval edge-colorings of bipartite graphs

“…By results of [16] and [18], all (2, b)-biregular graphs admit interval colorings (the latter result was also obtained independently by Kostochka [22]). In [10] it is proved that every (3,6)-biregular graph has an interval 7-coloring and in [11] it was proved that large families of (3,5)-biregular graphs admit interval colorings. Several sufficient conditions for a (3,4)-biregular graph to admit an interval 6-coloring have been obtained [5,23,27]; however, it remains an open question whether all such graphs have interval colorings.…”

“…Trivially, every bipartite graph G with parts X and Y has an X-interval coloring with |E(G)| colors. We denote by χ ′ int (G, X) the smallest integer t such that there is an X-interval t-coloring of G. Note that, in general, the problem of computing χ ′ int (G, X) is N P-hard; this follows from the fact that determining whether a given (3,6)-biregular graph has an interval 6-coloring is N P-complete [2].…”