Interleaved adjoints on directed graphs

TL;DR

This paper introduces the concept of interleaved adjoints on directed graphs, providing bounds on their chromatic number, and explores implications for graph homomorphisms and conjectures in graph theory.

Contribution

It defines the k-th interlacing adjoint of a digraph, derives bounds on its chromatic number, and connects these results to homomorphisms and the weak Hedetniemi conjecture.

Findings

Derived tight bounds on chromatic numbers of interlaced adjoints of transitive tournaments.

Proved the existence of directed paths with specific algebraic lengths mapping into graphs with high chromatic number.

Connected the properties of interlaced adjoints to implications for the weak Hedetniemi conjecture.

Abstract

For an integer k >= 1, the k-th interlacing adjoint of a digraph G is the digraph i_k(G) with vertex-set V(G)^k, and arcs ((u_1, ..., u_k), (v_1, ..., v_k)) such that (u_i,v_i) \in A(G) for i = 1, ..., k and (v_i, u_{i+1}) \in A(G) for i = 1, ..., k-1. For every k we derive upper and lower bounds for the chromatic number of i_k(G) in terms of that of G. In particular, we find tight bounds on the chromatic number of interlacing adjoints of transitive tournaments. We use this result in conjunction with categorial properties of adjoint functors to derive the following consequence. For every integer ell, there exists a directed path Q_{\ell} of algebraic length ell which admits homomorphisms into every directed graph of chromatic number at least 4. We discuss a possible impact of this approach on the multifactor version of the weak Hedetniemi conjecture.