In this paper we formulate the Vertex Congestion Lemma leading to a new technique in computing the exact wirelength of an embedding. We compute the circular wirelength of generalized Petersen graphs by partitioning the vertices as well as the edges of cycles. Further we obtain the linear wirelength of circular ladders. Our algorithms produce optimal values in linear time.

let $g$ be a simple graph of order $n$ and size $m$.the edge covering of $g$ is a set of edges such that every vertex of $g$ is incident to at least one edge of the set. the edge cover polynomial of $g$ is the polynomial$e(g,x)=sum_{i=rho(g)}^{m} e(g,i) x^{i}$,where $e(g,i)$ is the number of edge coverings of $g$ of size $i$, and$rho(g)$ is the edge covering number of $g$. in this paper we stud...

suppose $gamma$ is a graph with $v(gamma) = { 1,2, cdots, p}$and $ mathcal{f} = {gamma_1,cdots, gamma_p} $ is a family ofgraphs such that $n_j = |v(gamma_j)|$, $1 leq j leq p$. define$lambda = gamma[gamma_1,cdots, gamma_p]$ to be a graph withvertex set $ v(lambda)=bigcup_{j=1}^pv(gamma_j)$ and edge set$e(lambda)=big(bigcup_{j=1}^pe(gamma_j)big)cupbig(bigcup_{ijine(gamma)}{uv;uin v(gamma_i),vin ...

In the game of cops and robbers on a graph G = (V,E), k cops try to catch a robber. On the cop turn, each cop may move to a neighboring vertex or remain in place. On the robber’s turn, he moves similarly. The cops win if there is some time at which a cop is at the same vertex as the robber. Otherwise, the robber wins. The minimum number of cops required to catch the robber is called the cop num...

Petersen-Torus as an interconnection network was extensively studied. This letter proposed two elementary algorithms for QoS routing in general Petersen-Torus networks, which can be modeled as graph ( , ) PT m n with K QoS parameters associated with each edge. Theoretic analysis for the proposed algorithms was conducted. And the time complexity of these two algorithms was summarized and compare...

We use graph homomorphisms and the chromatic properties of the Petersen graph to prove some inequalities on systems of sets. These inequalities are then used to find counterexamples to a conjecture of Albertson and Collins on the monotonicity of the chromatic difference sequence of a vertex-transitive graph.

A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3} with the properties that if f(u)=0, then vertex u adjacent to at least one assigned 3 or two vertices 2, and f(u)=1, 2 3. The weight of f equals w(f)=?v?Vf(v). domination number ?dR(G) G minimum G. said be ?dR(G)=3?(G), where ?(G) We obtain sharp lower bound generalized Petersen graphs P(3k,k), we construct solutions providi...