Riis [Electron. J. Combin., 14(1):R44, 2007] introduced a guessing game for graphs which is equivalent to finding protocols for network coding. In this paper we prove upper and lower bounds for the winning probability of the guessing game on undirected graphs. We find optimal bounds for perfect graphs and minimally imperfect graphs, and present a conjecture relating the exact value for all grap...

A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph G is the maximum density of an independent set in G. Lih, Liu, and Zhu [31] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well stud...

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. As a main result, we compute the parameter for infinitely many graphs based on hypercubes. These graphs ...

A generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the colors on adjacent vertices are required to be as far apart as possible.

Let FG(P ) be a functional defined on the set of all the probability distributions on the vertex set of a graph G. We say that G is symmetric with respect to FG(P ) if the uniform distribution on V (G) maximizes FG(P ). Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic numb...

‎Let $f$ be a proper $k$-coloring of a connected graph $G$ and‎ ‎$Pi=(V_1,V_2,ldots,V_k)$ be an ordered partition of $V(G)$ into‎ ‎the resulting color classes‎. ‎For a vertex $v$ of $G$‎, ‎the color‎ ‎code of $v$ with respect to $Pi$ is defined to be the ordered‎ ‎$k$-tuple $c_{{}_Pi}(v)=(d(v,V_1),d(v,V_2),ldots,d(v,V_k))$‎, ‎where $d(v,V_i)=min{d(v,x):~xin V_i}‎, ‎1leq ileq k$‎. ‎If‎ ‎distinct...

Given a finite set D of positive integers, the distance graph G(Z,D) has Z as the vertex set and {ij : |i−j| ∈ D} as the edge set. Given D, the asymptotic clique covering ratio is defined as S(D) = lim sup n→∞ n cl(n) , where cl(n) is the minimum number of cliques covering any consecutive n vertices of G(Z,D). The parameter S(D) is closely related to the ratio spT(G) χ(G) of a graph G, where χ(...

This report gather some notable results found in [SS12], giving also details on some foundation results presented in [CMN+07]. The results covered in Section 2 are answers given in [SS12] to questions of the article [CMN+07]. Graph parameters are studied a lot as they are a way to highlight a general structure from a graph. In that sense, the chromatic number is studied a lot as it shows the mu...

An adjacent vertex distinguishing edge-coloring or an avd-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree ∆ and with no isolated edges has an avd-coloring with at most ∆ + 300 colors, provided that ∆ > 1020. AMS Subject Classification: 05C15

Let χc(H) denote the circular chromatic number of a graph H. For graphs F and G, the circular chromatic Ramsey number Rχc(F,G) is the infimum of χc(H) over graphs H such that every red/blue edge-coloring of H contains a red copy of F or a blue copy of G. We characterize Rχc(F,G) in terms of a Ramsey problem for the families of homomorphic images of F and G. Letting zk = 3 − 2 −k, we prove that ...