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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 fractional colouring of a graph $G$ is a function that assigns a non-negative real value to all possible colour-classes of $G$ containing any vertex of $G$, such that the sum of these values is at least one for each vertex. The fractional chromatic number is the minimum sum of the values assigned by a fractional colouring over all possible such colourings of $G$. Introduced by Bosica and Tardif, fractional covers are an extension of fractional colourings whereby the real-valued function acts on all possible subgraphs of $G$ belonging to a given class of graphs. The fractional chromatic number turns out to be a special instance of the fractional cover number. In this work we investigate fractional covers acting on $(k+1)$-clique-free subgraphs of $G$ which, although sharing some similarities with fractional covers acting on $k$-colourable subgraphs of $G$, they exhibit some peculiarities. We first show that if a simple graph $G_2$ is a homomorphic image of a simple graph $G_1$, then the fractional cover number defined on the $(k+1)$-clique-free subgraphs of $G_1$ is bounded above by the corresponding number of $G_2$. We make use of this result to obtain bounds for the associated fractional cover number of graphs that are either $n$-colourable or $a\!\!:\!\!b$-colourable., 8 pages

A pairing of a graph $G$ is a perfect matching of the complete graph having the same vertex set as $G$. If every pairing of $G$ can be extended to a Hamiltonian cycle of the underlying complete graph using only edges from $G$, then $G$ has the PH-property. A somewhat weaker property is the PMH-property, whereby every perfect matching of $G$ can be extended to a Hamiltonian cycle of $G$. In an attempt to characterise all 4-regular graphs having the PH-property, we answer a question made in 2015 by Alahmadi et al. by showing that the Cartesian product $C_p\square C_q$ of two cycles on $p$ and $q$ vertices does not have the PMH-property, except for $C_4\square C_4$ which is known to have the PH-property., 6 pages, 2 figures

In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic graph has three perfect matchings whose intersection is empty. In this paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a possibly weaker conjecture originally proposed by the first author, which states that in every simple bridgeless cubic graph there exist two perfect matchings such that the complement of their union is a bipartite graph. Here, we show that this conjecture can be equivalently stated using a variant of Petersen-colourings, we prove it for graphs having oddness at most four and we give a natural extension to bridgeless cubic multigraphs and to certain cubic graphs having bridges., 17 pages, 11 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