Linear Index Coding via Graph Homomorphism

It is known that the minimum broadcast rate of a linear index code over $\mathbb{F}_q$ is equal to the $minrank_q$ of the underlying digraph. In [3] it is proved that for $\mathbb{F}_2$ and any positive integer $k$, $minrank_q(G)\leq k$ iff there exists a homomorphism from the complement of the graph $G$ to the complement of a particular undirected graph family called "graph family $\{G_k\}$". As observed in [2], by combining these two results one can relate the linear index coding problem of undirected graphs to the graph homomorphism problem. In [4], a direct connection between linear index coding problem and graph homomorphism problem is introduced. In contrast to the former approach, the direct connection holds for digraphs as well and applies to any field size. More precisely, in [4], a graph family $\{H_k^q\}$ is introduced and shown that whether or not the scalar linear index of a digraph $G$ is less than or equal to $k$ is equivalent to the existence of a graph homomorphism from the complement of $G$ to the complement of $H_k^q$. Here, we first study the structure of the digraphs $H_k^q$. Analogous to the result of [2] about undirected graphs, we prove that $H_k^q$'s are vertex transitive digraphs. Using this, and by applying a lemma of Hell and Nesetril [5], we derive a class of necessary conditions for digraphs $G$ to satisfy $lind_q(G)\leq k$. Particularly, we obtain new lower bounds on $lind_q(G)$. Our next result is about the computational complexity of scalar linear index of a digraph. It is known that deciding whether the scalar linear index of an undirected graph is equal to $k$ or not is NP-complete for $k\ge 3$ and is polynomially decidable for $k=1,2$ [3]. For digraphs, it is shown in [6] that for the binary alphabet, the decision problem for $k=2$ is NP-complete. We use graph homomorphism framework to extend this result to arbitrary alphabet.