Induced matchings in cubic graphs

Abstract

In this paper, we show that the edge set of a cubic graph can always be partitioned into 10 subsets, each of which induces a matching in the graph. This result is a special case of a general conjecture made by Erdos and NeSetiil: For each d 2 3, the edge set of a graph of maximum degree d can always be partitioned into [5d2/4] subsets each of which induces a matching. 0 1993 John Wiley & Sons, Inc. INTRODUCTIONThroughout this paper, we consider colorings of the edges of a graph with positive integers. Formally, a t-coloring of a graph G = ( V , E ) is a map $: -{1,2,. . . , t}. A t-coloring is proper if $(e) = $(f) and e # f imply that the edges e and f have no common endpoints. Of course, the chromatic index of a graph G is the least t for which G has a proper tcoloring. Note that whenever qj is a proper t-coloring of a graph G = ( V , E ) and a E {1,2,. . . , t}, then the edges in 34 = {e E E:$(e) = a } form a matching in G.An induced matching 34 in a graph G = ( V , E ) is a matching such that no two edges of 34 are joined by an edge of G. In other words, an induced matching is an induced subgraph in which every vertex has degree one. A

Induced Matchings in Subcubic Planar Graphs

“…(Observe that the line graph of a planar graph with maximum degree 3 is a planar graph with maximum degree 4.) Independently, Andersen [1] and Horák, He, and Trotter [8] demonstrated that every subcubic graph has a strong edge 10-coloring, which implies that every subcubic graph with m edges has an induced matching of size at least m/10.…”

On isomorphic linear partitions in cubic graphs

“…When dealing with cubic graphs, we have immediately that χ S (G) ≥ 5. We know that χ S (G) ≤ 10 (see [4] and [13]) for cubic graphs in general and χ S (G) ≤ 9 (see [15]) when considering cubic bipartite graphs (answering thus positively to conjectures appearing in [9] and [8]). …”

On strong edge-colouring of subcubic graphs

“…The conjecture was verified for graphs having ∆ ≤ 3 [1,12]. When ∆ > 3, the only case on which some progress was made is when ∆ = 4 and the best upper bound stated is χ ′ s (G) ≤ 22 [4].…”