Progress on the adjacent vertex distinguishing edge colouring conjecture

A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\Delta$ and no isolated edge has an adjacent vertex distinguishing edge colouring with $\Delta + 300$ colours, provided $\Delta$ is large enough. We show that this bound can be reduced to $\Delta + 19$. This is motivated by the conjecture of Zhang, Liu, and Wang (2002) that $\Delta + 2$ colours are enough for $\Delta \geq 3$.Comment: v2: Revised following referees' comment

Rotulações graciosas e rotulações semifortes em grafos

Orientador: Christiane Neme CamposTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Três problemas de rotulação em grafos são investigados nesta tese: a Conjetura das Árvores Graciosas, a Conjetura 1,2,3 e a Conjetura 1,2. Uma rotulação graciosa de um grafo simples G=(V(G),E(G)) é uma função injetora f de V(G) em {0,...,|E(G)|} tal que {|f(u)-f(v)|: uv em E(G)} = {1,...,|E(G)|}. A Conjetura das Árvores Graciosas, proposta por Rosa e Kotzig em 1967, afirma que toda árvore possui uma rotulação graciosa. Um problema relacionado à Conjetura das Árvores Graciosas consiste em determinar se, para todo vértice v de uma árvore T, existe uma rotulação graciosa de T que atribui o rótulo 0 a v. Árvores com tal propriedade são denominadas 0-rotativas. Nesta tese, apresentamos famílias infinitas de caterpillars 0-rotativos. Nossos resultados reforçam a conjetura de que todo caterpillar com diâmetro pelo menos cinco é 0-rotativo. Também investigamos uma rotulação graciosa mais restrita, chamada rotulação-alpha. Uma rotulação graciosa f de G é uma rotulação-alpha se existir um inteiro k, 0 <= k <= |E(G)|, tal que, para toda aresta uv em E(G), f(u) <= k < f(v) ou f(v) <= k < f(u). Nesta tese, apresentamos duas famílias de lobsters com grau máximo três que possuem rotulações-alpha. Nossos resultados contribuem para uma caracterização de todos os lobsters com grau máximo três que possuem rotulações-alpha. Na segunda parte desta tese, investigamos generalizações da Conjetura 1,2,3 e da Conjetura 1,2. Dado um grafo simples G = (V(G),E(G)) e um subconjunto L dos números reais, dizemos que uma função f de E(G) em L é uma L-rotulação de arestas de G e dizemos que uma função f da união de V(G) com E(G) em L é uma L-rotulação total de G. Para todo vértice v de G, a cor de v, C(v), é definida como a soma dos rótulos das arestas incidentes em v, se f for uma L-rotulação de arestas de G. Se f for uma L-rotulação total, C(v) é a soma dos rótulos das arestas incidentes no vértice v mais o valor f(v). O par (f,C) é uma L-rotulação de arestas semiforte (L-rotulação total semiforte) se f for uma rotulação de arestas (rotulação total) e C(u) for diferente de C(v) para quaisquer dois vértices adjacentes u,v de G. A Conjetura 1,2,3, proposta por Karónski et al. em 2004, afirma que todo grafo simples e conexo com pelo menos três vértices possui uma {1,2,3}-rotulação de arestas semiforte. A Conjetura 1,2, proposta por Przybylo e Wozniak em 2010, afirma que todo grafo simples possui uma {1,2}-rotulação total semiforte. Sejam a,b,c três reais distintos. Nesta tese, nós investigamos {a,b,c}-rotulações de arestas semifortes e {a,b}-rotulações totais semifortes para cinco famílias de grafos: as potências de caminho, as potências de ciclo, os grafos split, os grafos cobipartidos regulares e os grafos multipartidos completos. Provamos que essas famílias possuem tais rotulações para alguns valores reais a,b,c. Como corolário de nossos resultados, obtemos que a Conjetura 1,2,3 e a Conjetura 1,2 são verdadeiras para essas famílias. Além disso, também mostramos que nossos resultados em rotulações de arestas semifortes implicam resultados similares para outro problema de rotulação de arestas relacionadoAbstract: This thesis addresses three labelling problems on graphs: the Graceful Tree Conjecture, the 1,2,3-Conjecture, and the 1,2-Conjecture. A graceful labelling of a simple graph G=(V(G),E(G)) is an injective function f from V(G) to {0,...,|E(G)|} such that {|f(u)-f(v)| : uv in E(G)} = {1,...,|E(G)|}. The Graceful Tree Conjecture, posed by Rosa and Kotzig in 1967, states that every tree has a graceful labelling. A problem connected with the Graceful Tree Conjecture consists of determining whether, for every vertex v of a tree T, there exists a graceful labelling of T that assigns label 0 to v. Trees with such a property are called 0-rotatable. In this thesis, we present infinite families of 0-rotatable caterpillars. Our results reinforce a conjecture that states that every caterpillar with diameter at least five is 0-rotatable. We also investigate a stronger type of graceful labelling, called alpha-labelling. A graceful labelling f of G is an alpha-labelling if there exists an integer k with 0<= k <= |E(G)| such that, for each edge uv in E(G), either f(u) <= k < f(v) or f(v) <= k < f(u). In this thesis, we prove that the following families of lobsters have alpha-labellings: lobsters with maximum degree three, without Y-legs and with at most one forbidden ending; and lobsters T with a perfect matching M such that the contracted tree T/M has a balanced bipartition. These results point towards a characterization of all lobsters with maximum degree three that have alpha-labellings. In the second part of the thesis, we focus on generalizations of the 1,2,3-Conjecture and the 1,2-Conjecture. Given a simple graph G=(V(G),E(G)) and a subset L of real numbers, we call a function f from E(G) to L an L-edge-labelling of G, and we call a function f from V(G) union E(G) to L an L-total-labelling of G. For each vertex v of G, the colour of v, C(v), is defined as the sum of the labels of its incident edges, if f is an L-edge-labelling. If f is an L-total-labelling, C(v) is the sum of the labels of the edges incident with vertex v plus the label f(v). The pair (f,C) is a neighbour-distinguishing L-edge-labelling (neighbour-distinguishing L-total-labelling) if f is an edge-labelling (total-labelling) and C(u) is different from C(v), for every edge uv in E(G). The 1,2,3-Conjecture, posed by Kar\'onski et al. in 2004, states that every connected simple graph with at least three vertices has a neighbour-distinguishing {1,2,3}-edge-labelling. The 1,2-Conjecture, posed by Przybylo and Wozniak in 2010, states that every simple graph has a neighbour-distinguishing {1,2}-total-labelling. Let a,b,c be distinct real numbers. In this thesis, we investigate neighbour-distinguishing {a,b,c}-edge-labellings and neighbour-distinguishing {a,b}-total labellings for five families of graphs: powers of paths, powers of cycles, split graphs, regular cobipartite graphs and complete multipartite graphs. We prove that these families have such labellings for some real values a, b, and c. As a corollary of our results, we obtain that the 1,2,3-Conjecture and the 1,2-Conjecture are true for these families. Furthermore, we also show that our results on neighbour-distinguishing edge-labellings imply similar results on a closely related problem called detectable edge-labelling of graphsDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação2014/16861-8FAPESPCAPE

On {a,b}-edge-weightings of bipartite graphs with odd a,b

International audienceFor any S⊂ℤ we say that a graph G has the S-property if there exists an S-edge-weighting w:E(G)→S such that for any pair of adjacent vertices u,v we have Σ_{e∈E(v)} w(e) ≠ Σ_{e∈E(u)} w(e), where E(v) and E(u) are the sets of edges incident to v and u respectively. This work focuses on {a,a+2}-edge-weightings where a∈ℤ is odd. We show that a 2-connected bipartite graph has the {a,a+2}-property if and only if it is not a so-called odd multi-cactus. In the case of trees, we show that only one case is pathological. That is, we show that all trees have the {a,a+2}-property for odd a≠−1, while there is an easy characterization of trees without the {−1,1}-property

On topological relaxations of chromatic conjectures

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number

A general decomposition theory for the 1-2-3 Conjecture and locally irregular decompositions

International audienceHow can one distinguish the adjacent vertices of a graph through an edge-weighting? In the last decades, this question has been attracting increasing attention, which resulted in the active field of distinguishing labellings. One of its most popular problems is the one where neighbours must be distinguishable via their incident sums of weights. An edge-weighting verifying this is said neighbour-sum-distinguishing. The popularity of this notion arises from two reasons. A first one is that designing a neighbour-sum-distinguishing edge-weighting showed up to be equivalent to turning a simple graph into a locally irregular (i.e., without neighbours with the same degree) multigraph by adding parallel edges, which is motivated by the concept of irregularity in graphs. Another source of popularity is probably the influence of the famous 1-2-3 Conjecture, which claims that such weightings with weights in {1,2,3} exist for graphs with no isolated edge. The 1-2-3 Conjecture has recently been investigated from a decompositional angle, via so-called locally irregular decompositions, which are edge-partitions into locally irregular subgraphs. Through several recent studies, it was shown that this concept is quite related to the 1-2-3 Conjecture. However, the full connexion between all those concepts was not clear. In this work, we propose an approach that generalizes all concepts above, involving coloured weights and sums. As a consequence, we get another interpretation of several existing results related to the 1-2-3 Conjecture. We also come up with new related conjectures, to which we give some support

Probabilistic Problems in Graph Theory

In this thesis, I examine two different problems in graph theory using probabilistic techniques. The first is a question on graph colourings. A proper total k-colouring of a graph G = (V, E) is a map φ : V υ E → {1, 2,…, k} such that φ|V is a proper vertex colouring, φ|E is a proper edge colouring, and if v V and vw E then φ(v) ≠ φ(vw). Such a colouring is called adjacent vertex distinguishing if for every pair of adjacent vertices, u and v, the set {φ(u)} υ {φ(uw) : uw E}, the `colour set of u\u27, is distinct from the colour set of v. It is shown that there is an absolute constant C such that the minimal number of colours needed for such a colouring is at most Δ(G) + C. The second problem is related to a modification of bootstrap percolation on a finite square grid. In an n × n grid, the 1 × 1 squares, called sites, can be in one of two states: `uninfected\u27 or `infected\u27. Sites are initially infected independently at random and the state of each vertex is updated simultaneously by the following rule: every uninfected site that shares an edge with at least two infected sites becomes itself infected while each infected site with no infected neighbours becomes uninfected. This process is repeated and the central question is, when is it either likely or unlikely that all sites eventually become infected? Here, both upper and lower bounds are given for the probability that all sites eventually become infected and these bounds are used to determine a critical probability for the event that all sites eventually become infected