On S-packing edge-colorings of cubic graphs

Abstract

Given a non-decreasing sequence S = (s 1 , s 2 , . . . , s k ) of positive integers, an Spacking edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 , X 2 , . . . , X k } such that for each 1 ≤ i ≤ k, the distance between two distinct edges e, e ′ ∈ X i is at least s i + 1. This paper studies S-packing edgecolorings of cubic graphs. Among other results, we prove that cubic graphs having a 2-factor are (1, 1, 1, 3, 3)-packing edge-colorable, (1, 1, 1, 4, 4, 4, 4, 4)-packing edgecolorable and (1, 1, 2, 2, 2, 2, 2)-packing edge-colorable. We determine sharper results for cubic graphs of bounded oddness and 3-edge-colorable cubic graphs and we propose many open problems.

Packing colorings of subcubic outerplanar graphs

“…Fiala and Golovach have shown that the decision version of the packing chromatic number is NP-complete even in the class of trees [12]. Packing coloring of some other classes of graphs, such as the distance graphs [11,22,25], hypercubes [26], subdivision graphs of subcubic graphs [4,10,15], and still other classes of graphs [2,18,20] was also studied.…”

“…On the other hand, the question whether in subcubic graphs the packing chromatic number is bounded was much more intriguing. It was posed in the seminal paper [17], and then investigated in several papers [8,9,15] using different approaches. Recently, Balogh, Kostochka and Liu [3] have provided a negative answer to the question.…”

Packing Chromatic Number of Subdivisions of Cubic Graphs

“…The concept has attracted a considerable attention recently: there are around 30 papers on the topic (see e.g. [1,3,4,5,6,7,8,9,10,11,12,13,22] and references in them). In particular, Fiala and Golovach [10] proved that finding the packing chromatic number of a graph is NP-hard even in the class of trees.…”

“…The questions on how large can χ p (G) and χ p (D(G)) be if G is a subcubic graph (i.e., a graph with maximum degree at most 3) were discussed in several papers (see [6,7,13,21,22]). In particular, Gastineau and Togni [13] asked whether χ p (D(G)) ≤ 5 for every subcubic graph G. Brešar, Klavžar, Rall, and Wash [7] later conjectured this and proved the validity of their conjecture for some special classes of subcubic graphs (e.g., the class of generalized Petersen graph). However, no upper bounds for the whole class of (sub)cubic graphs were proved in either case.…”

“…, k)-coloring is a packing k-coloring. For subcubic graphs, Gastineau and Togni [13] proved that they are (1, 1, 2, 2, 2)colorable and (1, 2, 2, 2, 2, 2, 2)-colorable. We will use the following observation of Gastineau and Togni [13].…”