Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

Abstract

Abstract:In this paper, we show that if G is a 3-edge-connected graph with S V ðGÞ and jSj 12, then either G has an Eulerian subgraph H such that S V ðHÞ, or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3-edge-connected planar graph, then for any jSj 23, G has an Eulerian subgraph H such that S V ðHÞ. As an application, we obtain a new result on Hamiltonian line graphs.

Hamiltonicity in 3-connected claw-free graphs

“…By Lemma 2.5 and by the fact that G is triangle-free, K is collapsible. By Claim 3 and Theorem 2.3, M 10 − D 1 (M 10 ) is collapsible, contrary to (5). So…”

“…Theorem 3.1. (Chen et al [5]) Let G be a 3-edge-connected graph and let S ⊆ V (G) be a vertex subset such that |S| 12. Then either G has an Eulerian subgraph C such that S ⊆ V (C), or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. [7]) Let S be a set of vertices of a graph G contained in an Eulerian subgraph of G and let C be a maximal Eulerian subgraph of G containing S. Assume that some component A of G − V (C) is not an isolated vertex and is related to C by at least r edges.…”

Supereulerian graphs and the Petersen graph

“…Chen et al [10,12,13] investigated conditions under which a 3-edge-connected graph G is supereulerian if and only if G cannot be contracted to the Petersen graph. These settled the 3-edge-connected case of a conjecture by Benhocine et al [1].…”

“…As |P ∪ Q| = |P | + |Q| − |P ∩ Q| = 8 < 9 = |S 1 | and by (4.4), |S 1 − P ∪ Q| = 1. If w 3 ∈ P ∪ Q, then by {w 10 , w 11 , w 12 , w 13 …”

Parameterized Edge Hamiltonicity

“…[5,6,9,7,23,8]). In this paper we investigate the complexity of this problem from the parameterized complexity perspective.…”