On Index Coding and Graph Homomorphism
Javad B. Ebrahimi, Mahdi Jafari Siavoshani

TL;DR
This paper explores the relationship between index coding and graph homomorphism, providing bounds and classifications that unify and extend previous results in the field.
Contribution
It introduces a graph homomorphism-based framework to analyze and bound index coding rates, unifying various existing bounds and extending results to directed graphs.
Findings
Upper bounds on index coding rates via graph homomorphisms
Classification of graphs based on linear index and homomorphisms
Lower bounds on scalar index using chromatic number
Abstract
In this work, we study the problem of index coding from graph homomorphism perspective. We show that the minimum broadcast rate of an index coding problem for different variations of the problem such as non-linear, scalar, and vector index code, can be upper bounded by the minimum broadcast rate of another index coding problem when there exists a homomorphism from the complement of the side information graph of the first problem to that of the second problem. As a result, we show that several upper bounds on scalar and vector index code problem are special cases of one of our main theorems. For the linear scalar index coding problem, it has been shown in [1] that the binary linear index of a graph is equal to a graph theoretical parameter called minrank of the graph. For undirected graphs, in [2] it is shown that if and only if there exists a homomorphism…
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