Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.Comment: 19 pages, 1 figur

Local Graph Coloring and Index Coding

We present a novel upper bound for the optimal index coding rate. Our bound uses a graph theoretic quantity called the local chromatic number. We show how a good local coloring can be used to create a good index code. The local coloring is used as an alignment guide to assign index coding vectors from a general position MDS code. We further show that a natural LP relaxation yields an even stronger index code. Our bounds provably outperform the state of the art on index coding but at most by a constant factor.Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013; typos correcte

Local chromatic number and Sperner capacity

We introduce a directed analog of the local chromatic number defined by Erdos et al. [Discrete Math. 59 (1986) 21-34] and show that it provides an upper bound for the Sperner capacity of a directed graph. Applications and variants of this result are presented. In particular, we find a special orientation of an odd cycle and show that it achieves the maximum of Sperner capacity among the differently oriented versions of the cycle. We show that apart from this orientation, for all the others an odd cycle has the same Sperner capacity as a single edge graph. We also show that the (undirected) local chromatic number is bounded from below by the fractional chromatic number while for power graphs the two invariants have the same exponential asymptotics (under the co-normal product on which the definition of Sperner capacity is based). We strengthen our bound on Sperner capacity by introducing a fractional relaxation of our directed variant of the local chromatic number. (C) 2005 Elsevier Inc. All rights reserved

Relations between the local chromatic number and its directed version

The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here, we consider oriented graphs. We show the existence of a graph where the directed local chromatic number of all oriented versions of the graph is strictly less than the local chromatic number of the underlying undirected graph. We show that for fractional versions the analogous problem has a different answer: there always exists an orientation for which the directed and undirected values coincide. We also determine the supremum of the possible ratios of these fractional parameters, which turns out to be e, the basis of the natural logarithm