Decomposing claw-free subcubic graphs and 4-chordal subcubic graphs

Abstract

Hoffmann-Ostenhof's Conjecture states that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a 2-regular subgraph. In this paper, we show that the conjecture holds for claw-free subcubic graphs and 4-chordal subcubic graphs. induced subgraph isomorphic to K 1,3 . A cycle is called chordless if it has no chord. A graph G is called chordal if every cycle of G of length greater than 3 has a chord and a graph is 4-chordal if it has no induced cycle of length greater than 4. A cut-edge of a connected graph G is an edge e ∈ E(G) such that G \ e is disconnected. A subdivision of a graph G is a graph obtained from G by replacing some of the edges of G by internally vertex-disjoint paths.Hoffmann-Ostenhof proposed the following conjecture in his thesis [5], this conjecture also was appeared as a problem of BCC22 [4].Hoffmann-Ostenhof 's Conjecture. Let G be a connected cubic graph. Then the edges of G can be decomposed into a spanning tree, a matching and a 2-regular subgraph.An edge decomposition of a graph G is called a good decomposition, if the edges of G can be decomposed into a spanning tree, a matching and a 2-regular subgraph (matching or 2-regular subgraph may be empty). A graph is called good if it has a good decomposition. Hoffmann-Ostenhof's Conjecture is known to be true for some families of cubic graphs. Kostochka [7] showed that the Petersen graph, the prism over cycles, and many other graphs are good.Bachstein [3] proved that every 3-connected cubic graph embedded in torus or Klein-bottle is good. Furthermore, Ozeki and Ye [8] proved that 3-connected cubic plane graphs are good.Akbari et. al. [2] showed that hamiltonian cubic graphs are good. Also, it has been proved that the traceable cubic graphs are good [1]. In 2017, Hoffmann-Ostenhof et. al. [6] proved that planar cubic graphs are good. In this paper it is shown that the connected claw-free subcubic graphs and the connected 4-chordal subcubic graphs are good. Connected Claw-free Subcubic Graphs are GoodIn this section we show that in the case of claw-free graphs, Hoffmann-Ostenhof's Conjecture holds even for claw-free subcubic graphs.Theorem 2.1. If G is a connected claw-free subcubic graph, then G is good.Proof. We apply by induction on n = |V (G)|. If |V (G)| = 3, then the assertion is trivial. Now, we consider the following two cases:Case 1. Assume that the graph G has a cut-edge e. Let H and K be the connected components of G \ e. By induction hypothesis, both H and K are good. Let T i , i = 1, 2, be

The 2-Decomposition Conjecture for a new class of graphs

“…We recall that a graph is claw-free if it contains no induced copy of K 1,3 . As mentioned before, the 3DC has been verified for claw-free cubic graphs [1,15]. Using an idea similar to the one used by Aboomahigir, Ahanjideh and Akbari [1], one can show that the 2DC also holds for claw-free graphs.…”

“…As mentioned before, the 3DC has been verified for claw-free cubic graphs [1,15]. Using an idea similar to the one used by Aboomahigir, Ahanjideh and Akbari [1], one can show that the 2DC also holds for claw-free graphs. For completeness, we provide its proof below.…”

“…It is straightforward from the assignments in steps (a)-(d) that F is a forest and M is a matching (for the edges that were subdivided, item (1) with respect to M ′ and Claim 2 ensure that M is a matching and that F has no cycles). We now check that items (1) and (2) hold for (F, ∈ I.…”

Towards Obtaining a 3-Decomposition From a Perfect Matching

“…The former of these results was extended to all connected plane cubic graphs by Hoffmann-Ostenhof, Kaiser, and Ozeki [8] in 2018. In the same year it was also proved to hold for claw-free (sub)cubic graphs by Aboomahigir, Ahanjideh, and Akbari [9] as well as by Hong et al [10]. More recently, Lyngsie and Merker [11] showed that weakening the matching requirement to allow for paths of length 2 suffices to make the conjecture true and Heinrich [12] verified it for 3-connected cubic graphs of tree-width 3.…”

Towards obtaining a 3-Decomposition from a perfect Matching

“…In the same year it was also proved to hold for claw-free (sub)cubic graphs by Aboomahigir, Ahanjideh, and Akbari [8]. More recently, Lyngsie and Merker [9] showed that weakening the matching requirement to allow for paths of length 2 suffices to make the conjecture true and Heinrich [10] proved the conjecture for 3-connected cubic graphs of tree-width 3.…”