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Let $G$ be a bridgeless cubic graph. The Berge--Fulkerson Conjecture (1970s) states that $G$ admits a list of six perfect matchings such that each edge of $G$ belongs to exactly two of these perfect matchings. If answered in the affirmative, two other recent conjectures would also be true: the Fan--Raspaud Conjecture (1994), which states that $G$ admits three perfect matchings such that every edge of $G$ belongs to at most two of them; and a conjecture by Mazzuoccolo (2013), which states that $G$ admits two perfect matchings whose deletion yields a bipartite subgraph of $G$. It can be shown that given an arbitrary perfect matching of $G$, it is not always possible to extend it to a list of three or six perfect matchings satisfying the statements of the Fan--Raspaud and the Berge--Fulkerson conjectures, respectively. In this paper, we show that given any $1^+$-factor $F$ (a spanning subgraph of $G$ such that its vertices have degree at least 1) and an arbitrary edge $e$ of $G$, there always exists a perfect matching $M$ of $G$ containing $e$ such that $G\setminus (F\cup M)$ is bipartite. Our result implies Mazzuoccolo's conjecture, but not only. It also implies that given any collection of disjoint odd circuits in $G$, there exists a perfect matching of $G$ containing at least one edge of each circuit in this collection., 13 pages, 8 figures

Let $G$ be a graph of even order and let $K_{G}$ be the complete graph on the same vertex set of $G$. A pairing of a graph $G$ is a perfect matching of the graph $K_{G}$. A graph $G$ has the Pairing-Hamiltonian property (for short, the PH-property) if for each one of its pairings, there exists a perfect matching of $G$ such that the union of the two gives rise to a Hamiltonian cycle of $K_G$. In 2015, Alahmadi \emph{et al.} gave a complete characterisation of the cubic graphs having the PH-property. Most naturally, the next step is to characterise the quartic graphs that have the PH-property. In this work we propose a class of quartic graphs on two parameters, $n$ and $k$, which we call the class of accordion graphs $A[n,k]$. We show that an infinite family of quartic graphs (which are also circulant) that Alahmadi \emph{et al.} stated to have the PH-property are, in fact, members of this general class of accordion graphs. We also study the PH-property of this class of accordion graphs, at times considering the pairings of $G$ which are also perfect matchings of $G$. Furthermore, there is a close relationship between accordion graphs and the Cartesian product of two cycles. Motivated by a recent work by Bogdanowicz (2015), we give a complete characterisation of those accordion graphs that are circulant graphs. In fact, we show that $A[n,k]$ is not circulant if and only if both $n$ and $k$ are even, such that $k\geq 4$., 18 pages, 9 figures

A graph $G$ has the Perfect-Matching-Hamiltonian property (PMH-property) if for each one of its perfect matchings, there is another perfect matching of $G$ such that the union of the two perfect matchings yields a Hamiltonian cycle of $G$. The study of graphs that have the PMH-property, initiated in the 1970s by Las Vergnas and H\"{a}ggkvist, combines three well-studied properties of graphs, namely matchings, Hamiltonicity and edge-colourings. In this work, we study these concepts for cubic graphs in an attempt to characterise those cubic graphs for which every perfect matching corresponds to one of the colours of a proper 3-edge-colouring of the graph. We discuss that this is equivalent to saying that such graphs are even-2-factorable (E2F), that is, all 2-factors of the graph contain only even cycles. The case for bipartite cubic graphs is trivial, since if $G$ is bipartite then it is E2F. Thus, we restrict our attention to non-bipartite cubic graphs. A sufficient, but not necessary, condition for a cubic graph to be E2F is that it has the PMH-property. The aim of this work is to introduce an infinite family of E2F non-bipartite cubic graphs on two parameters, which we coin papillon graphs, and determine the values of the respective parameters for which these graphs have the PMH-property or are just E2F. We also show that no two papillon graphs with different parameters are isomorphic., Comment: 18 pages, 11 figures

A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a 2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that every bridgeless cubic graph, apart from the well-known Petersen graph, admits a 2-bisection. In the same paper it was shown that every Class I bridgeless cubic graph admits such a bisection. The Class II bridgeless cubic graphs which are critical to many conjectures in graph theory are known as snarks, in particular, those with excessive index at least 5, that is, whose edge set cannot be covered by four perfect matchings. Moreover, in [J. Graph Theory, 86(2) (2017), 149--158], Esperet et al. state that a possible counterexample to Ban--Linial's Conjecture must have circular flow number at least 5. The same authors also state that although empirical evidence shows that several graphs obtained from the Petersen graph admit a 2-bisection, they can offer nothing in the direction of a general proof. Despite some sporadic computational results, until now, no general result about snarks having excessive index and circular flow number both at least 5 has been proven. In this work we show that treelike snarks, which are an infinite family of snarks heavily depending on the Petersen graph and with both their circular flow number and excessive index at least 5, admit a 2-bisection., Comment: 10 pages, 6 figures

A Hamiltonian graph is 2-factor Hamiltonian (2FH) if each of its 2-factors is a Hamiltonian cycle. A similar, but weaker, property is the Perfect-Matching-Hamiltonian property (PMH-property): a graph admitting a perfect matching is said to have this property if each one of its perfect matchings (1-factors) can be extended to a Hamiltonian cycle. It was shown that the star product operation between two bipartite 2FH-graphs is necessary and sufficient for a bipartite graph admitting a 3-edge-cut to be 2FH. The same cannot be said when dealing with the PMH-property, and in this work we discuss how one can use star products to obtain graphs (which are not necessarily bipartite, regular and 2FH) admitting the PMH-property with the help of malleable vertices, which we introduce here. We show that the presence of a malleable vertex in a graph implies that the graph has the PMH-property, but does not necessarily imply that it is 2FH. It was also conjectured that if a graph is a bipartite cubic 2FH-graph, then it can only be obtained from the complete bipartite graph $K_{3,3}$ and the Heawood graph by using star products. Here, we show that a cubic graph (not necessarily bipartite) is 2FH if and only if all of its vertices are malleable. We also prove that the above conjecture is equivalent to saying that, apart from the Heawood graph, every bipartite cyclically 4-edge-connected cubic graph with girth at least 6 having the PMH-property admits a perfect matching which can be extended to a Hamiltonian cycle in exactly one way. Finally, we also give two necessary and sufficient conditions for a graph admitting a 2-edge-cut to be: (i) 2FH, and (ii) PMH., Comment: 19 pages, 15 figures

A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a Hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH-property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most $3$, (ii) a complete graph, or (iii) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper., Comment: 12 pages, 4 figures