Graphs with large palette index

Given an edge-coloring of a graph, the palette of a vertex is defined as the set of colors of the edges which are incident with it. We define the palette index of a graph as the minimum number of distinct palettes, taken over all edge-colorings, occurring among the vertices of the graph. Several results about the palette index of some specific classes of graphs are known. In this paper we propose a different approach that leads to new and more general results on the palette index. Our main theorem gives a sufficient condition for a graph to have palette index larger than its minimum degree. In the second part of the paper, by using such a result, we answer to two open problems on this topic. First, for every $r$ odd, we construct a family of $r$-regular graphs with palette index reaching the maximum admissible value. After that, we construct the first known family of simple graphs whose palette index grows quadratically with respect to their maximum degree.Comment: 7 pages, 1 figur

Pairwise disjoint perfect matchings in rr-edge-connected rr-regular graphs

Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every $r$-edge-connected $r$-regular graph of even order has $r-2$ pairwise disjoint perfect matchings. We show that this is not the case if $r \equiv 2 \text{ mod } 4$. Together with a recent result of Mattiolo and Steffen [Highly edge-connected regular graphs without large factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves Thomassen's problem for all even $r$. It turns out that our methods are limited to the even case of Thomassen's problem. We then prove some equivalences of statements on pairwise disjoint perfect matchings in highly edge-connected regular graphs, where the perfect matchings contain or avoid fixed sets of edges. Based on these results we relate statements on pairwise disjoint perfect matchings of 5-edge-connected 5-regular graphs to well-known conjectures for cubic graphs, such as the Fan-Raspaud Conjecture, the Berge-Fulkerson Conjecture and the $5$-Cycle Double Cover Conjecture.Comment: 24 page

Nowhere-zero Circular Flows e Fattori di Grafi - Costruzioni e Controesempi

Uno dei campi di ricerca di grande interesse nella teoria dei grafi strutturale è quello dei nowhere-zero integer flows. Una delle ragioni principali è il fatto che questi oggetti generalizzano il concetto di colorazione propria sulle facce di grafi planari. Si ricordi il Teorema dei 4-Colori del 1976, uno dei teoremi più famosi della teoria dei grafi, che afferma che ogni grafo planare senza ponti ammette una 4-colorazione propria sulle facce. Nel 1954, Tutte dimostrò che un grafo planare ammette una k-colorazione propria sulle facce se e solo se ammette un nowhere-zero k-flow e congetturò che ogni grafo senza ponti ammette un nowhere-zero 5-flow. Questa congettura, conosciuta come la Congettura dei 5-Flussi di Tutte, è uno dei problemi aperti più importanti in quest'area della matematica. È ben noto che la Congettura sui 5-Flussi è equivalente alla sua restrizione agli snarks, grafi cubici che non ammettono una 3-colorazione propria sugli spigoli e con ulteriori restrizioni sulla cintura e connettività ciclica. Molte altre congetture importanti possono essere ridotte alla classe degli snarks e questo è uno dei motivi principali per cui questi grafi sono studiati in molti articoli. In questa tesi studiamo i nowhere-zero circular flows, che sono una generalizzazione dei nowhere-zero integer flows, e, in particolare, studiamo il numero di flusso circolare di grafi. Infatti, negli ultimi decenni, i nowhere-zero circular flows hanno attratto molti autori ed ora, in questo campo, si trovano parecchi problemi di ricerca e congetture aperte. I risultati presentati in questa tesi sono motivati da queste nuove congetture, che in alcuni casi risolviamo in modo parziale o completo. La maggior parte di tali risultati sono costruzioni di famiglie infinite di snarks e, più in generale, grafi con determinate proprietà strutturali. Sottolineiamo che alcune famiglie presentate sono i primi esempi noti aventi determinate caratteristiche mentre altre contengono controesempi a problemi aperti.Nowhere-zero integer flows in graphs represent a research field of great interest in structural graph theory. One of the main reasons is the fact that they generalize the concept of face-coloring of planar graphs. Recall that one of the most famous theorems of graph theory is for sure the 4-Color Theorem (1976), claiming that every bridgeless planar graph admits a proper face-4-coloring. In 1954, Tutte proved that a planar graph has a proper face-k-coloring if and only if it has a nowhere-zero k-flow and conjectured that every bridgeless graph admits a nowhere-zero 5-flow. This conjecture is known as the 5-Flow Conjecture and is one of the most important and outstanding open problems in this area of mathematics. It is well known that the 5-Flow Conjecture is equivalent to its restriction to snarks, that are non-3-edge-colorable cubic graphs with further technical requirements on the girth and cyclic edge-connectivity. Many other important long-standing conjectures can be reduced to the family of snarks and this is the main reason why snarks are studied in many papers. In this dissertation, we study nowhere-zero circular flows, that are a generalization of nowhere-zero integer flows, and, in particular, we study the circular flow number of graphs. Indeed, in the last decades, circular flows have attracted many authors and now some problems and conjectures are left open in this field. The results that appear in the present thesis are motivated by these new problems and conjectures, which in some cases we partially or fully answer to. Most of such results are constructions of infinite families of snarks and, more generally, graphs having specific structural properties. We remark that a few of such families are the first known examples having certain properties, and others contain counterexamples to open research problems

On 3-Bisections in Cubic and Subcubic Graphs

A k-bisection of a graph is a partition of the vertices in two classes whose cardinalities differ of at most one and such that the subgraphs induced by each class are acyclic with all connected components of order at most k. Esperet, Tarsi and the second author proved in 2017 that every simple cubic graph admits a 3-bisection. Recently, Cui and Liu extended that result to the class of simple subcubic graphs. Their proof is an adaptation of the quite long proof of the cubic case to the subcubic one. Here, we propose an easier proof of a slightly stronger result. Indeed, starting from the result for simple cubic graphs, we prove the existence of a 3-bisection for all cubic graphs (also admitting parallel edges). Then we prove the same result for the larger class of subcubic graphs as an easy corollary

A unified approach to construct snarks with circular flow number 5

The well-known 5-flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5-flow Conjecture can be reduced to cubic graphs, in particular to snarks. However, very few procedures to construct snarks with circular flow number 5 are known. In the first part of this paper, we summarise some of these methods and we propose new ones based on variations of the known constructions. Afterwards, we prove that all such methods are nothing but particular instances of a more general construction that we introduce into detail. In the second part, we consider many instances of this general method and we determine when our method permits to obtain a snark with circular flow number 5. Finally, by a computer search, we determine all snarks having circular flow number 5 up to 36 vertices. It turns out that all such snarks of order at most 34 can be obtained by using our method, and that the same holds for 96 of the 98 snarks of order 36 with circular flow number 5

Computational results and new bounds for the circular flow number of snarks

It is well known that the circular flow number of a bridgeless cubic graph can be computed in terms of certain partitions of its vertex set with prescribed properties. In the present paper, we first study some of these properties that turn out to be useful in order to make an efficient and practical implementation of an algorithm for the computation of the circular flow number of a bridgeless cubic graph. Using this procedure, we determine the circular flow number of all snarks on up to 36 vertices as well as the circular flow number of various famous snarks. After that, as combination of the use of this algorithm with new theoretical results, we present an infinite family of snarks of order 8k+2 whose circular flow numbers meet a general lower bound presented by Lukot'ka and Skoviera in 2008. In particular this answers a question proposed in their paper. Moreover, we improve the best known upper bound for the circular flow number of Goldberg snarks and we conjecture that this new upper bound is optimal. (C) 2020 Elsevier B.V. All rights reserved