On linear index coding from graph homomorphism perspective

In this work, we study the problem of linear index coding from graph homomorphism point of view. We show that the decision version of linear (scalar or vector) index coding problem is equivalent to certain graph homomorphism problem. Using this equivalence expression, we conclude the following results. First we introduce new lower bounds on linear index of graphs. Next, we show that if the linear index of a graph over a finite filed is bounded by a constant, then by changing the ground field, the linear index of the graph may change by at most a constant factor that is independent from the size of the graph. Finally, we show that the decision version of linear index coding problem is NP-Complete unless we want to decide if this quantity is equal to 1 in which case the problem is solvable in polynomial time.