Consider the following game on a graph $G$: Alice and Bob take turns coloring the vertices of $G$ properly from a fixed set of colors; Alice wins when the entire graph has been colored, while Bob wins when some uncolored vertices have been left. The game chromatic number of $G$ is the minimum number of colors that allows Alice to win the game. The game Grundy number of $G$ is defined similarly except that the players color the vertices according to the first-fit rule and they only decide on the order in which it is applied. The $(a,b)$-game chromatic and Grundy numbers are defined likewise except that Alice colors $a$ vertices and Bob colors $b$ vertices in each round. We study the behavior of these parameters for incomparability graphs of posets with bounded width. We conjecture a complete characterization of the pairs $(a,b)$ for which the $(a,b)$-game chromatic and Grundy numbers are bounded in terms of the width of the poset; we prove that it gives a necessary condition and provide some evidence for its sufficiency. We also show that the game chromatic number is not bounded in terms of the Grundy number, which answers a question of Havet and Zhu.
Despite the fact that some vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not "under control" from a polyhedral point of view. The equivalence between \emph{optimization} and \emph{polyhedral separation} suggests that, for these problems, there must exist formulations admitting some elegant characterization for the polytopes associated to them. Therefore, it is interesting to study known formulations for vertex coloring with the goal of finding such characterizations. In this work we study the asymmetric representatives formulation and w…
The distinguishing index $D'(Γ)$ of a graph $Γ$ is the least number $k$ such that $Γ$ has an edge-coloring with $k$ colors preserved only by the trivial automorphism. In this paper we prove that if the automorphism group of a finite graph $Γ$ is simple, then its distinguishing index $D'(Γ)=2$.
We show that the edges of every 3-connected planar graph except $K_4$ can be colored with two colors in such a way that the graph has no color preserving automorphisms. Also, we characterize all graphs which have the property that their edges can be $2$-colored so that no matter how the graph is embedded in any orientable surface, there is no homeomorphism of the surface which induces a non-trivial color preserving automorphism of the graph.
An edge-coloring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Lehner, Pilśniak, and Stawiski proved that all connected regular graphs except $K_2$ admit an asymmetric edge-coloring with three colors. We generalize this result for graphs whose minimal degree $δ$ and the maximal degree $Δ$ satisfy $δ\geq Δ/2$.
Topological Coding consists of two different kinds of mathematics: topological structure and mathematical relation. The colorings and labelings of graph theory are main techniques in topological coding applied in asymmetric encryption system. Topsnut-gpws (also, colored graphs) have the following advantages: (1) Run fast in communication networks because they are saved in computer by popular matrices rather than pictures. (2) Produce easily text-based (number-based) strings for encrypt files. (3) Diversity of asymmetric ciphers, one public-key corresponds to more private-keys, or more public-k…
A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex $x$ is called a spectrum of $x$. An arbitrary nonempty subset of consecutive integers is called an interval. Suppose that all edges of a graph $G$ are colored in the game of Alice and Bob with asymmetric distribution of roles. Alice determines the number $t$ of colors in the future proper edge coloring of $G$ and aspires to minimize the number of vertices with an interva…
Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such that for any red-blue-green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of at least size $l$ in a component, or a green matching of size at least $m$ in a component. In this paper, we determine the exact value of $r(k,l,m)$ completely. Applying a technique originated by Łuczak that applies Szemerédi's Regularity Lemma to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component, we obtain th…
Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge probability $p$) satisfies with probability $1-o(1)$ that every red-blue-coloring of its edges contains a red copy of $G$ or a blue copy of $H$. We prove a general upper bound on the threshold for this property under the assumption that the denser of the two graphs satisfies a certain balancedness condition. Our result partially confirms a conjecture by the first author and Kreuter, and together with earlier lower bound results establishes th…
A graph $F$ is Ramsey for a pair of graphs $(G,H)$ if any red/blue-coloring of the edges of $F$ yields a copy of $G$ with all edges colored red or a copy of $H$ with all edges colored blue. Two pairs of graphs are called Ramsey equivalent if they have the same collection of Ramsey graphs. The symmetric setting, that is, the case $G=H$, received considerable attention. This led to the open question whether there are connected graphs $G$ and $G'$ such that $(G,G)$ and $(G',G')$ are Ramsey equivalent. We make progress on the asymmetric version of this question and identify several non-trivial fam…
A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such a coloring, and the distinguishing threshold of $G$, denoted by $θ(G)$, is the minimum number $k$ such that every $k$-coloring of $G$ is distinguishing. As an alternative definition, $θ(G)$ is one more than the maximum number of cycles in the cycle decomposition of automorphisms of $G$. In this paper, we characterize $θ(G)$ when $G$ is disconnected. Afterwards, we prove that, alt…
