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arXiv · 2211.12813

On the expansion constant and distance constrained colourings of hypergraphs

Annayat Ali·Rameez Raja
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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.

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Annayat Ali, Rameez Raja. 2022-11-23. On the expansion constant and distance constrained colourings of hypergraphs. https://arxiv.org/abs/2211.12813

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