A new upper bound on the game chromatic index of graphs

We study the two-player game where Maker and Breaker alternately color the edges of a given graph $G$ with $k$ colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, all edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored edge where every color is blocked. The game chromatic index $\chi'_g(G)$ denotes the smallest $k$ for which Maker has a winning strategy. The trivial bounds $\Delta(G) \le \chi_g'(G) \le 2\Delta(G)-1$ hold for every graph $G$, where $\Delta(G)$ is the maximum degree of $G$. In 2008, Beveridge, Bohman, Frieze, and Pikhurko proved that for every $\delta>0$ there exists a constant $c>0$ such that $\chi'_g(G) \le (2-c)\Delta(G)$ holds for any graph with $\Delta(G) \ge (\frac{1}{2}+\delta)v(G)$, and conjectured that the same holds for every graph $G$. In this paper, we show that $\chi'_g(G) \le (2-c)\Delta(G)$ is true for all graphs $G$ with $\Delta(G) \ge C \log v(G)$. In addition, we consider a biased version of the game where Breaker is allowed to color $b$ edges per turn and give bounds on the number of colors needed for Maker to win this biased game.Comment: 17 page

Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Mini-Workshop: Positional Games

Positional games is one of rapidly developing subjects of modern combinatorics, researching two player perfect information games of combinatorial nature, ranging from recreational games like Tic-Tac-Toe to purely abstract games played on graphs and hypergraphs. Though defined usually in game theoretic terms, the subject has a distinct combinatorial flavor and boasts strong mutual connections with discrete probability, Ramsey theory and randomized algorithms. This mini-workshop was dedicated to summarizing the recent progress in the subject, to indicating possible directions of future developments, and to fostering collaboration between researchers working in various, sometimes apparently distinct directions

Line game-perfect graphs

The $[X,Y]$-edge colouring game is played with a set of $k$ colours on a graph $G$ with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player $X\in\{A,B\}$ has the first move. $Y\in\{A,B,-\}$. If $Y\in\{A,B\}$, then only player $Y$ may skip any move, otherwise skipping is not allowed for any player. A move consists in colouring an uncoloured edge with one of the $k$ colours such that adjacent edges have distinct colours. When no more moves are possible, the game ends. If every edge is coloured in the end, Alice wins; otherwise, Bob wins. The $[X,Y]$-game chromatic index $\chi_{[X,Y]}'(G)$ is the smallest nonnegative integer $k$ such that Alice has a winning strategy for the $[X,Y]$-edge colouring game played on $G$ with $k$ colours. The graph $G$ is called line $[X,Y]$-perfect if, for any edge-induced subgraph $H$ of $G$, $$\chi_{[X,Y]}'(H)=\omega(L(H)),$$ where $\omega(L(H))$ denotes the clique number of the line graph of $H$. For each of the six possibilities $(X,Y)\in\{A,B\}\times\{A,B,-\}$, we characterise line $[X,Y]$-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively

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

Identification, estimating and testing in the restricted linear model

Estimation;Testing;62J05;62K99;statistics