Structural Properties of Index Coding Capacity Using Fractional Graph Theory

The capacity region of the index coding problem is characterized through the notion of confusion graph and its fractional chromatic number. Based on this multiletter characterization, several structural properties of the capacity region are established, some of which are already noted by Tahmasbi, Shahrasbi, and Gohari, but proved here with simple and more direct graph-theoretic arguments. In particular, the capacity region of a given index coding problem is shown to be simple functionals of the capacity regions of smaller subproblems when the interaction between the subproblems is none, one-way, or complete.Comment: 5 pages, to appear in the 2015 IEEE International Symposium on Information Theory (ISIT

Nonrepetitive colorings of lexicographic product of graphs

A coloring $c$ of the vertices of a graph $G$ is nonrepetitive if there exists no path $v_1v_2\ldots v_{2l}$ for which $c(v_i)=c(v_{l+i})$ for all $1\le i\le l$. Given graphs $G$ and $H$ with $|V(H)|=k$, the lexicographic product $G[H]$ is the graph obtained by substituting every vertex of $G$ by a copy of $H$, and every edge of $G$ by a copy of $K_{k,k}$. %Our main results are the following. We prove that for a sufficiently long path $P$, a nonrepetitive coloring of $P[K_k]$ needs at least $3k+\lfloor k/2\rfloor$ colors. If $k>2$ then we need exactly $2k+1$ colors to nonrepetitively color $P[E_k]$, where $E_k$ is the empty graph on $k$ vertices. If we further require that every copy of $E_k$ be rainbow-colored and the path $P$ is sufficiently long, then the smallest number of colors needed for $P[E_k]$ is at least $3k+1$ and at most $3k+\lceil k/2\rceil$. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results

Digraphs and homomorphisms: Cores, colorings, and constructions

A natural digraph analogue of the graph-theoretic concept of an `independent set\u27 is that of an acyclic set, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets

A unified construction of semiring-homomorphic graph invariants

It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a new characterization of the Shannon capacity $\Theta$ via Strassen's Positivstellensatz: $\Theta(\bar{G}) = \inf_f f(G)$, where $f : \mathsf{Graph} \to \mathbb{R}_+$ ranges over all monotone semiring homomorphisms. Constructing and classifying graph invariants $\mathsf{Graph} \to \mathbb{R}_+$ which are monotone under graph homomorphisms, additive under join, and multiplicative under disjunctive product is therefore of major interest. We call such invariants semiring-homomorphic. The only known such invariants are all of a fractional nature: the fractional chromatic number, the projective rank, the fractional Haemers bounds, as well as the Lov\'asz number (with the latter two evaluated on the complementary graph). Here, we provide a unified construction of these invariants based on linear-like semiring families of graphs. Along the way, we also investigate the additional algebraic structure on the semiring of graphs corresponding to fractionalization. Linear-like semiring families of graphs are a new concept of combinatorial geometry different from matroids which may be of independent interest.Comment: 25 pages. v3: incorporated referee's suggestion