arXiv subjects

Gábor Simonyi

Publications and source records attributed to Gábor Simonyi.

22 records · Page 2Linked to original sources

arXiv2007

On types of growth for graph-different permutations

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity "within a given type".

math.CO↗

arXiv2007

Graph-different permutations

We strengthen and put in a broader perspective previous results of the first two authors on colliding permutations. The key to the present approach is a new non-asymptotic invariant for graphs.

math.CO↗

arXiv2007

Local chromatic number and distinguishing the strength of topological obstructions

The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K_2,G) and its suspension, respectively. These investigations follow the line of research initiated by Matousek and Ziegler who recognized a hierarchy of the different topological expressions that can serve as lower bounds for the chromatic number of a graph. Our results imply that the local chromatic number of 4-chromatic Kneser, Schrijver, Borsuk, and generalized Mycielski graphs is 4, and more generally, that 2r-chromatic versions of these graphs have local chromatic number at least r+2. This lower bound is tight in several cases by results in an earlier paper of the first two authors.

math.CO↗

arXiv2005

Colorful subgraphs in Kneser-like graphs

Combining Ky Fan's theorem with ideas of Greene and Matousek we prove a generalization of Dol'nikov's theorem. Using another variant of the Borsuk-Ulam theorem due to Bacon and Tucker, we also prove the presence of all possible completely multicolored t-vertex complete bipartite graphs in t-colored t-chromatic Kneser graphs and in several of their relatives. In particular, this implies a generalization of a recent result of G. Spencer and F. E. Su.

math.CO↗