Disproving the normal graph conjecture

A graph $G$ is called normal if there exist two coverings, $\mathbb{C}$ and $\mathbb{S}$ of its vertex set such that every member of $\mathbb{C}$ induces a clique in $G$, every member of $\mathbb{S}$ induces an independent set in $G$ and $C \cap S \neq \emptyset$ for every $C \in \mathbb{C}$ and $S \in \mathbb{S}$. It has been conjectured by De Simone and K\"orner in 1999 that a graph $G$ is normal if $G$ does not contain $C_5$, $C_7$ and $\overline{C_7}$ as an induced subgraph. We disprove this conjecture

Complementary Graph Entropy, AND Product, and Disjoint Union of Graphs

In the zero-error Slepian-Wolf source coding problem, the optimal rate is given by the complementary graph entropy $\overline{H}$ of the characteristic graph. It has no single-letter formula, except for perfect graphs, for the pentagon graph with uniform distribution $G_5$, and for their disjoint union. We consider two particular instances, where the characteristic graphs respectively write as an AND product $\wedge$, and as a disjoint union $\sqcup$. We derive a structural result that equates $\overline{H}(\wedge \: \cdot)$ and $\overline{H}(\sqcup \: \cdot)$ up to a multiplicative constant, which has two consequences. First, we prove that the cases where $\overline{H}(\wedge \:\cdot)$ and $\overline{H}(\sqcup \: \cdot)$ can be linearized coincide. Second, we determine $\overline{H}$ in cases where it was unknown: products of perfect graphs; and $G_5 \wedge G$ when $G$ is a perfect graph, using Tuncel et al.'s result for $\overline{H}(G_5 \sqcup G)$. The graphs in these cases are not perfect in general

Entropy and Graphs

The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This graph functional originated from the problem of source coding in information theory and was introduced by J. K\"{o}rner in 1973. Although the notion of graph entropy has its roots in information theory, it was proved to be closely related to some classical and frequently studied graph theoretic concepts. For example, it provides an equivalent definition for a graph to be perfect and it can also be applied to obtain lower bounds in graph covering problems. In this thesis, we review and investigate three equivalent definitions of graph entropy and its basic properties. Minimum entropy colouring of a graph was proposed by N. Alon in 1996. We study minimum entropy colouring and its relation to graph entropy. We also discuss the relationship between the entropy and the fractional chromatic number of a graph which was already established in the literature. A graph $G$ is called \emph{symmetric with respect to a functional $F_G(P)$} defined on the set of all the probability distributions on its vertex set if the distribution $P^*$ maximizing $F_G(P)$ is uniform on $V(G)$. 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 number, we prove that vertex transitive graphs are symmetric with respect to graph entropy. Furthermore, we show that a bipartite graph is symmetric with respect to graph entropy if and only if it has a perfect matching. As a generalization of this result, we characterize some classes of symmetric perfect graphs with respect to graph entropy. Finally, we prove that the line graph of every bridgeless cubic graph is symmetric with respect to graph entropy.Comment: 89 pages, 4 figures, MMath Thesi