Ban--Linial's Conjecture and treelike snarks

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 snarks, in particular, those with excessive index at least 5, that is, whose edge-set cannot be covered by four perfect matchings. Moreover, 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 figure

Some snarks are worse than others

Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for cubic graphs. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless cubic graph which is not 3--edge-colourable. In this paper we deal with the fact that the family of potential counterexamples to many interesting conjectures can be narrowed even further to the family ${\cal S}_{\geq 5}$ of bridgeless cubic graphs whose edge set cannot be covered with four perfect matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover Conjecture and the Fan-Raspaud Conjecture are examples of statements for which ${\cal S}_{\geq 5}$ is crucial. In this paper, we study parameters which have the potential to further refine ${\cal S}_{\geq 5}$ and thus enlarge the set of cubic graphs for which the mentioned conjectures can be verified. We show that ${\cal S}_{\geq 5}$ can be naturally decomposed into subsets with increasing complexity, thereby producing a natural scale for proving these conjectures. More precisely, we consider the following parameters and questions: given a bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii) how many copies of the same perfect matching need to be added, and (iii) how many 2--factors need to be added so that the resulting regular graph is Class I? We present new results for these parameters and we also establish some strong relations between these problems and some long-standing conjectures.Comment: 27 pages, 16 figure

On a family of quartic graphs: Hamiltonicity, matchings and isomorphism with circulants

A pairing of a graph $G$ is a perfect matching of the underlying complete graph $K_{G}$. A graph $G$ has 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 et al. proved that the only three cubic graphs having the PH-property are the complete graph $K_{4}$, the complete bipartite graph $K_{3,3}$, and the $3$-dimensional cube $\mathcal{Q}_{3}$. Most naturally, the next step is to characterise the quartic graphs that have the PH-property, and the same authors mention that there exists an infinite family of quartic graphs (which are also circulant graphs) having 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 $A[n,k]$, and show that the quartic graphs having the PH-property mentioned by Alahmadi et al. 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 $k\geq 4$.Comment: 17 pages, 9 figure

Disjoint odd circuits in a bridgeless cubic graph can be quelled by a single perfect matching

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.Comment: 13 pages, 8 figure

A note on fractional covers of a graph

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.Comment: 8 page

The Erd\H{o}s--Faber--Lov\'{a}sz Conjecture revisited

The Erd\H{o}s--Faber--Lov\'{a}sz Conjecture, posed in 1972, states that if a graph $G$ is the union of $n$ cliques of order $n$ (referred to as defining $n$-cliques) such that two cliques can share at most one vertex, then the vertices of $G$ can be properly coloured using $n$ colours. Although still open after almost 50 years, it can be easily shown that the conjecture is true when every shared vertex belongs to exactly two defining $n$-cliques. We here provide a quick and easy algorithm to colour the vertices of $G$ in this case, and discuss connections with clique-decompositions and edge-colourings of graphs.Comment: 6 page

Three-cuts are a charm: acyclicity in 3-connected cubic graphs

Let $G$ be a bridgeless cubic graph. In 2023, the three authors solved a conjecture (also known as the $S_4$-Conjecture) made by Mazzuoccolo in 2013: there exist two perfect matchings of $G$ such that the complement of their union is a bipartite subgraph of $G$. They actually 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 exists a perfect matching $M$ of $G$ containing $e$ such that $G\setminus (F\cup M)$ is bipartite. This is a step closer to comprehend better the Fan--Raspaud Conjecture and eventually the Berge--Fulkerson Conjecture. The $S_4$-Conjecture, now a theorem, is also the weakest assertion in a series of three conjectures made by Mazzuoccolo in 2013, with the next stronger statement being: there exist two perfect matchings of $G$ such that the complement of their union is an acyclic subgraph of $G$. Unfortunately, this conjecture is not true: Jin, Steffen, and Mazzuoccolo later showed that there exists a counterexample admitting 2-cuts. Here we show that, despite of this, every cyclically 3-edge-connected cubic graph satisfies this second conjecture.Comment: 21 pages, 12 figures. arXiv admin note: text overlap with arXiv:2204.1002

Extending perfect matchings to Hamiltonian cycles in line graphs

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 figure

Sulle ineccepibili relazioni tra i matching perfetti

Snarks e Hamiltonicit\ue0 sono due rilevanti aree di ricerca nella Teoria dei Grafi. Come suggerisce il titolo, in questa tesi studieremo come interagiscono tra loro i matching perfetti di un grafo, in particolare ci occuperemo di studiare la loro unione e intersezione nei due ambiti sopra indicati. Gli snarks, grafi cubici privi di ponti e di Classe II, sono oggetti cruciali quando si considerano congetture su grafi cubici senza ponti, perch\ue8, tipicamente, se una congettura viene verificata per gli snarks allora \ue8 valida in generale. Una di queste congetture, nota per il suo enunciato particolarmente semplice ma completamente aperta dopo oltre mezzo secolo, \ue8 la Congettura di Berge\u2013Fulkerson. Nella prima parte di questa tesi studieremo altre due ben note congetture entrambe conseguenze della congettura di Berge\u2013Fulkerson e che sono anche esse ancora aperte: la Congettura di Fan\u2013Raspaud (Fan & Raspaud 1994) e la Congettura S4 (Mazzuoccolo 2013). La prima riguarda il comportamento dell\u2019intersezione di tre matching perfetti e l\u2019altra il complemento dell\u2019unione di due matching perfetti. Diamo una formulazione equivalente della Congettura di Fan\u2013Raspaud che in letteratura appariva essere una versione pi\uf9 forte, e mostriamo che la Congettura S4 \ue8 vera per grafi cubici senza ponti di oddness al pi\uf9 4. A causa degli ostacoli che si incontrano nello studiare tali tipi di problemi, sono stati fatti molti tentativi di colmare il gap tra grafi cubici di Classe I e di Classe II introducendo invarianti che misurino quanto un grafo di Classe II \ue8 lontano dall\u2019essere di Classe I. In questo spirito, proponiamo di considerare il minimo numero di matching perfetti (non necessariamente distinti) che \ue8 necessario aggiungere a un grafo cubico senza ponti per ottenere un multigrafo di Classe I. Dimostriamo che il grafo di Petersen \ue8 sostanzialmente l\u2019unica ostruzione per questo problema, nel senso che non ammette un numero finito di matching perfetti con la propriet\ue0 richiesta. Inoltre, colleghiamo lo studio di questo problema ad altri ben noti: l\u2019excessive index e la lunghezza della pi\uf9 corta copertura in cicli. Nella seconda parte della tesi, studiamo un problema legato all\u2019Hamiltonicit\ue0 di un grafo, gi\ue0 introdotto negli anni settanta da Las Vergnas e H\ue4ggkvist, e poi generalizzato pi\uf9 di recente da Fink (2007). Ci riferiamo a grafo Hamiltoniani con un numero pari di vertici (condizione necessaria per avere un matching perfetto): in tali grafi, un ciclo Hamiltoniano lo si pu\uf2 vedere come unione di due matching perfetti. Diremo che G ha la Perfect-Matching-Hamiltonian property (in breve la PMH-property) se ogni suo matching perfetto si pu\uf2 estendere a un ciclo Hamiltoniano. Una propriet\ue0 ancora pi\uf9 forte \ue8 la seguente: un grafo G ha la Pairing-Hamiltonian property (in breve la PH-property) se ogni pairing di G (cio\ue8 un matching perfetto del grafo completo definito sugli stessi vertici di G) pu\uf2 essere esteso a un ciclo Hamiltoniano del grafo completo soggiacente usando un matching perfetto di G. Una caratterizzazione dei grafi cubici con la PH-property \ue8 stata fornita da Alahmadi e al. (2015). Gli stessi autori hanno solo parazialmente tentato una caratterizzazione anche dei grafi 4-regolari con la stessa propriet\ue0. Noi risolviamo uno dei problemi da loro proposti e mostriamo una famiglia di grafi 4-regolari, che chiameremo accordion, che riteniamo interessante in quest\u2019ambito. Le propriet\ue0 di Hamiltonicit\ue0 sono state ampiamente studiate anche per i line-graphs da, tra gli altri, Kotzig (1964), Harary & Nash-Williams (1965) e Thomassen (1986). In questa linea di ricerca diamo condizioni sufficienti per un grafo che garantiscano la PMH-property per il suo line-graph. Otteniamo tali risultati per grafi di grado al pi\uf9 3, grafi completi e grafi arbitrarily traceable. Infine otteniamo una caratterizzazione completa dei line graphs dei grafi bipartiti completi che ammettono la PH-property.Snarks and Hamiltonicity: two prominent areas of research in graph theory. As the title of the thesis suggests, here we study how perfect matchings behave together, more precisely, their union and intersection, in each of these two settings. Snarks, which for us represent Class II bridgeless cubic graphs, are crucial when considering conjectures about bridgeless cubic graphs, and, if such statements are true for snarks then they would be true for all bridgeless cubic graphs. One such conjecture which is known for its simple statement, but still indomitable after half a century, is the Berge\u2013Fulkerson Conjecture. In the first part of the thesis we study two other related and well-known conjectures about bridgeless cubic graphs, both consequences of the Berge\u2013Fulkerson Conjecture which are still very much open: the Fan\u2013Raspaud Conjecture (Fan & Raspaud, 1994) and the S4-Conjecture (Mazzuoccolo, 2013), dealing with the intersection of three perfect matchings, and the complement of the union of two perfect matchings, respectively. In particular, we give an equivalent formulation of the Fan\u2013Raspaud Conjecture which at first glance seems stronger, and show that the S4-Conjecture is true for bridgeless cubic graphs having oddness at most 4. Given the obstacles encountered when dealing with such problems, many have considered trying to bridge the gap between Class I and Class II bridgeless cubic graphs by looking at invariants that measure how far Class II bridgeless cubic graphs are from being Class I. In this spirit we consider a parameter which gives the least number of perfect matchings (not necessarily distinct) needed to be added to a bridgeless cubic graph such that the resulting multigraph is Class I. We show that the Petersen graph is, in some sense, the only obstruction for a bridgeless cubic graph to have a finite value for the parameter studied. We also relate this parameter to already well-studied concepts: the excessive index, and the length of a shortest cycle cover of a bridgeless cubic graph. In the second part, we study a concept about Hamiltonicity first considered in the 1970s by Las Vergnas and H\ue4ggkvist, which was generalised and recently brought to the limelight again by Fink (2007). In this part we look at Hamiltonian cycles in graphs having an even order (a necessary condition for a graph to admit a perfect matching). In such graphs, a Hamiltonian cycle can be seen as the union of two perfect matchings. If every perfect matching of a graph G extends to a Hamiltonian cycle, we say that G has the Perfect-Matching-Hamiltonian property (for short the PMH-property). A stronger property than the PMH-property is the following: a graph G has the Pairing-Hamiltonian property (for short the PH-property) if every pairing of G (i.e. a perfect matching of the complete graph having the same vertex set as G) can be extended to a Hamiltonian cycle of the underlying complete graph by using a perfect matching of G. A characterisation of all the cubic graphs having the PH-property was done by Alahmadi et al. (2015), and the same authors attempt to answer a most natural question, that of characterising all 4-regular graphs having the same property. We solve a problem they suggest regarding quartic graphs and the above properties, and propose a class of quartic graphs on two parameters, accordion graphs, which we believe is a rich one in this sense. Hamiltonicity was also heavily studied with respect to line graphs by Kotzig (1964), Harary & Nash-Williams (1965), and Thomassen (1986), amongst others, and along the same lines, we give sufficient conditions for a graph in order to guarantee the PMH-property in its line graph. We do this for subcubic graphs, complete graphs, and arbitrary traceable graphs. A complete characterisation of which line graphs of complete bipartite graphs admit the PH-property is given