Proper path‐factors and interval edge‐coloring of (3,4)‐biregular bigraphs

Abstract

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3, 4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3, 4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide sufficient conditions for the existence of such a subgraph.

On Interval Edge Colorings of Biregular Bipartite Graphs With Small Vertex Degrees

“…i to z (1) i and z (2) i by two edges. A schematic of the graph L in the case when k = 3 appears in Figure 4.…”

“…Several sufficient conditions for a (3,4)-biregular graph to admit an interval coloring has been obtained [2,7,19,22]. In [6] we give a sufficient condition for a (3, 5)-biregular graph to admit an interval coloring.…”

One-sided interval edge-colorings of bipartite graphs

“…Nevertheless, trees [11,5], regular and complete bipartite graphs [11,5], grids [9], and simple outerplanar bipartite graphs [10,6] all have interval colorings. 2 A well-known conjecture suggests that all (a, b)-biregular graphs have interval colorings (see e.g. [11,14,19]), where a bipartite graph is (a, b)-biregular if all vertices in one part have degree a and all vertices in the other part have degree b.…”

“…By results of [11,13], all (2, b)-biregular graphs admit interval colorings (the latter result was also obtained independently by Kostochka [16] and by Kamalian et al 3 ). Several sufficient conditions for a (3,4)-biregular graph G to admit an interval 6-coloring have been obtained [2,17,20]; however, it is still open whether all (3,4)-biregular graphs have interval colorings. In [8] we proved that every (3,6)-biregular graph has an interval 7-coloring and in [7] it was proved that large families of (3, 5)-biregular graphs admit interval colorings.…”

Some results on cyclic interval edge colorings of graphs

“…Later the theory of interval colorings was developed in e.g. [3,4,[7][8][9][10][11][12][13][14][15][21][22][23]25,27,30]. Generally, it is an NP-complete problem to determine whether a bipartite graph has an interval coloring [27].…”