On the expansion constant and distance constrained colourings of hypergraphs

TL;DR

This paper investigates the properties of L(h, k)-colourings in hypergraphs, establishing bounds and relations with spectral gap, expansion, and other invariants, and explores colourings in hypertrees and hypergraph products.

Contribution

It introduces new bounds for L(h, k)-chromatic numbers in hypergraphs, relating them to spectral and structural invariants, and extends results to hypertrees and hypergraph Cartesian products.

Findings

Derived inequalities linking L(2, 1)-chromatic number to spectral gap and expansion constant.

Established sharp upper bounds for L(2, 1)-chromatic number of hypertrees.

Analyzed L(2, 1)-colouring in Cartesian products of hypergraphs.

Abstract

For any two non-negative integers h and k, h > k, an L(h, k)-colouring of a graph G is a colouring of vertices such that adjacent vertices admit colours that at least differ by h and vertices that are two distances apart admit colours that at least differ by k. The smallest positive integer {\delta} such that G permits an L(h, k)-colouring with maximum colour {\delta} is known as the L(h, k)-chromatic number (L(h, k)-colouring number) denoted by {\lambda}_{h,k}(G). In this paper, we discuss some interesting invariants in hypergraphs. In fact, we study the relation between the spectral gap and L(2, 1)-chromatic number of hypergraphs. We derive some inequalities which relates L(2, 1)-chromatic number of a k-regular simple graph to its spectral gap and expansion constant. The upper bound of L(h, k)-chromatic number in terms of various hypergraph invariants such as strong chromatic number, strong independent number and maximum degree is obtained. We determine the sharp upper bound for L(2, 1)-chromatic number of hypertrees in terms of its maximum degree. Finally, we conclude this paper with a discussion on L(2, 1)-colouring in cartesian product of some classes of hypergraphs.