The Relaxed Game Chromatic Index of \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs

The (r, d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r, d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with ∆(G) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆+k−1 and d≥2k2 + 4k

The Relaxed Edge-Coloring Game and \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with ∆(F) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆ − j and d ≥ 2j + 2 for 0 ≤ j ≤ ∆ − 1. This both improves and generalizes the result for trees in [10]. More broadly, we generalize the main result in [10] by showing that if G is k-degenerate with ∆(G) = ∆ and j ∈ [∆ + k − 1], then there exists a function h(k, j) such that Alice has a winning strategy for this game with r = ∆ + k − j and d ≥ h(k, j)

The game chromatic number of random graphs

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number \chi_g(G) is the minimum k for which the first player has a winning strategy. In this paper we analyze the asymptotic behavior of this parameter for a random graph G_{n,p}. We show that with high probability the game chromatic number of G_{n,p} is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs

Digraph Coloring Games and Game-Perfectness

In this thesis the game chromatic number of a digraph is introduced as a game-theoretic variant of the dichromatic number. This notion generalizes the well-known game chromatic number of a graph. An extended model also takes into account relaxed colorings and asymmetric move sequences. Game-perfectness is defined as a game-theoretic variant of perfectness of a graph, and is generalized to digraphs. We examine upper and lower bounds for the game chromatic number of several classes of digraphs. In the last part of the thesis, we characterize game-perfect digraphs with small clique number, and prove general results concerning game-perfectness. Some results are verified with the help of a computer program that is discussed in the appendix

Refined activation strategy for the marking game

AbstractThis paper introduces a new strategy for playing the marking game on graphs. Using this strategy, we prove that if G is a planar graph, then the game colouring number of G, and hence the game chromatic number of G, is at most 17

The game Grundy number of graphs

Given a graph G = (V;E), two players, Alice and Bob, alternate their turns in choosing uncoloured vertices to be coloured. Whenever an uncoloured vertex is chosen, it is coloured by the least positive integer not used by any of its coloured neighbours. Alice's goal is to minimize the total number of colours used in the game, and Bob's goal is to maximize it. The game Grundy number of G is the number of colours used in the game when both players use optimal strategies. It is proved in this paper that the maximum game Grundy number of forests is 3, and the game Grundy number of any partial 2-tree is at most 7

GG-Parking Functions, Acyclic Orientations and Spanning Trees

Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of {\em maximum} $G$-parking functions and provide a new bijection between them and the set of spanning trees of $G$ with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the discrete $n$-cube $Q_n$. We present the article in an expository self-contained form, since we found the combinatorial aspects of $G$-parking functions somewhat scattered in the literature, typically treated in conjunction with sandpile models and closely related chip-firing games.Comment: Added coauthor, extension of v2 with additional results and references. 28 pages, 2 figure

Positional Games

Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.Comment: Submitted to Proceedings of the ICM 201

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

Logical limit laws for minor-closed classes of graphs

Let $\mathcal G$ be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all {\em connected} graphs in $\mathcal G$ on $n$ vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in $\mathcal G$ on $n$ vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface $S$. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of length $\approx 5\cdot 10^{-6}$. Finally, we analyse examples of non-addable classes where the behaviour is quite different. For instance, the zero-one law does not hold for the random caterpillar on $n$ vertices, even in FO.Comment: minor changes; accepted for publication by JCT