2022/11/23 by Annayat Ali, Ali, Annayat, Rameez Raja +1
Engineering · Neuroscience · #Combinatorics (math.CO) #FOS: Mathematics #Nuclear Receptors and Signaling #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2211.12813
openalex publication_date 2022/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 δ such that G permits an L(h, k)-colouring with maximum colour δ is known as the L(h, k)-chromatic number (L(h, k)-colouring number) denoted by λh,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.