Choice Number and Energy of Graphs

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved that E(G)>= 2(n-\chi(\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>= 2ch(G) for all graphs G except for those in a few specified families, where \bar{G}, \chi(G), and ch(G) are the complement, the chromatic number, and the choice number of G, respectively.Comment: to appear in Linear Algebra and its Application

Jogos combinatórios em grafos: jogo Timber e jogo de Coloração

Studies three competitive combinatorial games. The timber game is played in digraphs, with each arc representing a domino, and the arc direction indicates the direction in which it can be toppled, causing a chain reaction. The player who topples the last domino is the winner. A P-position is an orientation of the edges of a graph in which the second player wins. If the graph has cycles, then the graph has no P-positions and, for this reason, timber game is only interesting when played in trees. We determine the number of P-positions in three caterpillar families and a lower bound for the number of P-positions in any caterpillar. Moreover, we prove that a tree has P-positions if, and only if, it has an even number of edges. In the coloring game, Alice and Bob take turns properly coloring the vertices of a graph, Alice trying to minimize the number of colors used, while Bob tries to maximize them. The game chromatic number is the smallest number of colors that ensures that the graph can be properly colored despite of Bob's intention. We determine the game chromatic number for three forest subclasses (composed by caterpillars), we present two su cient conditions and two necessary conditions for any caterpillar to have game chromatic number equal to 4. In the marking game, Alice and Bob take turns selecting the unselected vertices of a graph, and Alice tries to ensure that for some integer k, every unselected vertex has at most k − 1 neighbors selected. The game coloring number is the smallest k possible. We established lower and upper bounds for the Nordhaus-Gaddum type inequality for the number of P-positions of a caterpillar, the game chromatic and coloring numbers in any graph.Estudo de três jogos combinatórios competitivos. O jogo timber é jogado em digrafos, sendo que cada arco representa um dominó, e o sentido do arco indica o sentido em que o mesmo pode ser derrubado, causando um efeito em cadeia. O jogador que derrubar o último dominó é o vencedor. Uma P-position é uma orientação das arestas de um grafo na qual o segundo jogador ganha. Se o grafo possui ciclos, então não há P-positions e, por este motivo, o jogo timber só é interessante quando jogado em árvores. Determinamos o número de P-positions em três famílias de caterpillars e um limite inferior para o número de P-positions em uma caterpillar qualquer. Além disto, provamos que uma árvore qualquer possui P-positions se, e somente se, possui quantidade par de arestas. No jogo de coloração, Alice e Bob se revezam colorindo propriamente os vértices de um grafo, sendo que Alice tenta minimizar o número de cores, enquanto Bob tenta maximizá-lo. O número cromático do jogo é o menor número de cores que garante que o grafo pode ser propriamente colorido apesar da intenção de Bob. Determinamos o número cromático do jogo para três subclasses de orestas (compostas por caterpillars), apresentamos duas condições su cientes e duas condições necessárias para qualquer caterpillar ter número cromático do jogo igual a 4. No jogo de marcação, Alice e Bob selecionam alternadamente os vértices não selecionados de um grafo, e Alice tenta garantir que para algum inteiro k, todo vértice não selecionado tem no máximo k − 1 vizinhos selecionados. O número de coloração do jogo é o menor k possível. Estabelecemos limites inferiores e superiores para a relação do tipo Nordhaus-Gaddum referente ao número de P-positions de uma caterpillar, aos números cromático e de coloração do jogo em um grafo qualquer

Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

On b-colorings and b-continuity of graphs

A b-coloring of G is a proper vertex coloring such that there is a vertex in each color class, which is adjacent to at least one vertex in every other color class. Such a vertex is called a color-dominating vertex. The b-chromatic number of G is the largest k such that there is a b-coloring of G by k colors. Moreover, if for every integer k, between chromatic number and b-chromatic number, there exists a b-coloring of G by k colors, then G is b-continuous. Determining the b-chromatic number of a graph G and the decision whether the given graph G is b-continuous or not is NP-hard. Therefore, it is interesting to find new results on b-colorings and b-continuity for special graphs. In this thesis, for several graph classes some exact values as well as bounds of the b-chromatic number were ascertained. Among all we considered graphs whose independence number, clique number, or minimum degree is close to its order as well as bipartite graphs. The investigation of bipartite graphs was based on considering of the so-called bicomplement which is used to determine the b-chromatic number of special bipartite graphs, in particular those whose bicomplement has a simple structure. Then we studied some graphs whose b-chromatic number is close to its t-degree. At last, the b-continuity of some graphs is studied, for example, for graphs whose b-chromatic number was already established in this thesis. In particular, we could prove that Halin graphs are b-continuous.:Contents 1 Introduction 2 Preliminaries 2.1 Basic terminology 2.2 Colorings of graphs 2.2.1 Vertex colorings 2.2.2 a-colorings 3 b-colorings 3.1 General bounds on the b-chromatic number 3.2 Exact values of the b-chromatic number for special graphs 3.2.1 Graphs with maximum degree at most 2 3.2.2 Graphs with independence number close to its order 3.2.3 Graphs with minimum degree close to its order 3.2.4 Graphs G with independence number plus clique number at most number of vertices 3.2.5 Further known results for special graphs 3.3 Bipartite graphs 3.3.1 General bounds on the b-chromatic number for bipartite graphs 3.3.2 The bicomplement 3.3.3 Bicomplements with simple structure 3.4 Graphs with b-chromatic number close to its t-degree 3.4.1 Regular graphs 3.4.2 Trees and Cacti 3.4.3 Halin graphs 4 b-continuity 4.1 b-spectrum of special graphs 4.2 b-continuous graph classes 4.2.1 Known b-continuous graph classes 4.2.2 Halin graphs 4.3 Further graph properties concerning b-colorings 4.3.1 b-monotonicity 4.3.2 b-perfectness 5 Conclusion Bibliograph

Distances and Domination in Graphs

This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs