On the expansion constant and distance constrained colourings of hypergraphs

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

Lambda number of zero-divisor graphs of finite commutative rings

Let \G = (\V, \E) be a simple graph, an $L(2,1)$-labelling of \G is assignment of labels from non-negative integers to the vertices of \G such that adjacent vertices gets labels which at least differ by two and vertices which are at distance two from each other get different labels. The $\lambda$-number of \G, denoted by \lambda(\G) is the smallest positive integer $\ell$ such that \G has a $L(2,1)$-labelling with all the labels are members of the set $\{ 0, 1, \cdots, \ell \}$. The zero-divisor graph denoted by $\Gamma(R)$, of a finite commutative ring $R$ with unity is a simple graph with vertices as non-zero zero divisors of $R$. Two vertices $u$ and $v$ are adjacent in $\Gamma(R)$ if and only if $uv = 0$ in $R$. In this paper, we investigate $L(2,1)$-labelling in zero-divisor graphs. We study the \textit{partite truncation}, a graph operation that reduces a $n$-partite graph of higher order to a graph of lower order. We establish the relation between $\lambda$-numbers of two graphs. We make use of the operation \textit{partite truncation} to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring. We compute the exact value of $\lambda$-numbers of zero-divisor graphs of some classes of local and mixed rings such as $\Z_{p^n}, \Z_{p^n} \times\Z_{q^m},$ and $\mathbb{F}_{q}\times\Z_{p^n}$