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Register a new userSome results on domination number of the graph defined by two levels of the n-cube
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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 that two vertices $S, epsilon ,{[n] choose k} $ and $T, epsilon ,{[n] choose l} $ are adjacent if and only if $T subset S$. In this paper, we give an upper bound for the domination number of graph $G_{k,2}$ for $k > lceil frac{n}{2} rceil$ and exact value for $k=n-1$.
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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 $gamma (G_{k,2})={k+3over 2(k-1)(k+1)}n^2+o(n^2)$.
The $k$-power domination problem is a problem in graph theory, which has applications in many areas. However, it is hard to calculate the exact $k$-power domination number since determining k-power domination number of a generic graph is a NP-complete problem. We determine the exact $k$-power domination number in two graphs which have the same number of vertices and edges: pseudofractal scale-free web and Sierpinski gasket. The $k$-power domination number becomes 1 for $kge2$ in the Sierpinski gasket, while the $k$-power domination number increases at an exponential rate with regard to the number of vertices in the pseudofractal scale-free web. The scale-free property may account for the difference in the behavior of two graphs.
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. Similarly, $D subset V(G)$ is a non-porous exponential dominating set is $1le sum_{d in D} left( frac{1}{2} right)^{overline{text{dist}}(d,v) -1}$ for every $v in V(G),$ where $overline{text{dist}}(d,v)$ represents the length of the shortest $dv$ path with no internal vertices in $D.$ The porous and non-porous exponential dominating number of $G,$ denoted $gamma_e^*(G)$ and $gamma_e(G),$ are the minimum cardinality of a porous and non-porous exponential dominating set, respectively. The consecutive circulant graph, $C_{n, [ell]},$ is the set of $n$ vertices such that vertex $v$ is adjacent to $v pm i mod n$ for each $i in [ell].$ In this paper we show $gamma_e(C_{n, [ell]}) = gamma_e^*(C_{n, [ell]}) = leftlceil tfrac{n}{3ell +1} rightrceil.$
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 that r(K_3,Q_n) = 2^{n+1} - 1 for every n in N, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K_3,Q_n) le 7000 cdot 2^n. Here we show that r(K_3,Q_n) = (1 + o(1)) 2^{n+1} as n to infty.
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 graphs, path graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the domination number of middle graphs.
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