An upper bound for the crossing number of augmented cubes

A {\it good drawing} of a graph $G$ is a drawing where the edges are non-self-intersecting and each two edges have at most one point in common, which is either a common end vertex or a crossing. The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a good drawing of $G$ in the plane. The {\it $n$-dimensional augmented cube} $AQ_n$, proposed by S.A. Choudum and V. Sunitha, is an important interconnection network with good topological properties and applications. In this paper, we obtain an upper bound on the crossing number of $AQ_n$ less than $26/324^{n}-(2n^2+7/2n-6)2^{n-2}$.Comment: 39 page

Roman domination number of Generalized Petersen Graphs P(n,2)

A $Roman\ domination\ function$ on a graph $G=(V, E)$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The $weight$ of a Roman domination function $f$ is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a Roman dominating function on a graph $G$ is called the $Roman\ domination\ number$ of $G$, denoted by $\gamma_{R}(G)$. In this paper, we study the {\it Roman domination number} of generalized Petersen graphs P(n,2) and prove that $\gamma_R(P(n,2)) = \lceil {\frac{8n}{7}}\rceil (n \geq 5)$.Comment: 9 page

The crossing number of locally twisted cubes

The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the crossing number of the hypercube. J. Graph Theory {\bf 59}, 145--161 (2008)] which solves the upper bound conjecture on the crossing number of $n$-dimensional hypercube proposed by Erd\H{o}s and Guy, we give upper and lower bounds of the crossing number of locally twisted cube, which is one of variants of hypercube.Comment: 17 pages, 12 figure