ﻻ يوجد ملخص باللغة العربية
Consider all $k$-element subsets and $ell$-element subsets $(k>ell )$ of an $n$-element set as vertices of a bipartite graph. Two vertices are adjacent if the corresponding $ell$-element set is a subset of the corresponding $k$-element set. Let $G_{k,ell}$ denote this graph. The domination number of $G_{k,1}$ was exactly determined by Badakhshian, Katona and Tuza. A conjecture was also stated there on the asymptotic value ($n$ tending to infinity) of the domination number of $G_{k,2}$. Here we prove the conjecture, determining the asymptotic value of the domination number $gamma (G_{k,2})={k+3over 2(k-1)(k+1)}n^2+o(n^2)$.
Let ${[n] choose k}$ and ${[n] choose l}$ $( k > l ) $ where $[n] = {1,2,3,...,n}$ denote the family of all $k$-element subsets and $l$-element subsets of $[n]$ respectively. Define a bipartite graph $G_{k,l} = ({[n] choose k},{[n] choose l},E)$ such
The Ramsey number r(K_3,Q_n) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K_N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and ErdH{o}s conjectured
For a graph $G,$ we consider $D subset V(G)$ to be a porous exponential dominating set if $1le sum_{d in D}$ $left( frac{1}{2} right)^{text{dist}(d,v) -1}$ for every $v in V(G),$ where dist$(d,v)$ denotes the length of the smallest $dv$ path. Similar
In this paper, we study the domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph. We also compute the domination number of some families of graphs such as star graphs, double start grap
The domination polynomials of binary graph operations, aside from union, join and corona, have not been widely studied. We compute and prove recurrence formulae and properties of the domination polynomials of families of graphs obtained by various pr