Induced path number for the complementary prism of a grid graph

The induced path number rho(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path. A complementary prism of a graph G that we will refer to as CP(G) is the graph formed from the disjoint union of G and G_bar and adding the edges between the corresponding vertices of G and G_bar. These new edges are called prism edges. The graph grid(n,m) is the Cartesian product of P_n with P_m. In this thesis we will give an overview of a selection of important results in determining rho(G) of various graphs, we will then provide proofs for determining the exact value of rho(CP(grid(n,m))) for specific values of n and m

A necessary and sufficient condition for lower bounds on crossing numbers of generalized periodic graphs in an arbitrary surface

Let $H$, $T$ and $C_n$ be a graph, a tree and a cycle of order $n$, respectively. Let $H^{(i)}$ be the complete join of $H$ and an empty graph on $i$ vertices. Then the Cartesian product $H\Box T$ of $H$ and $T$ can be obtained by applying zip product on $H^{(i)}$ and the graph produced by zip product repeatedly. Let $\textrm{cr}_{\Sigma}(H)$ denote the crossing number of $H$ in an arbitrary surface $\Sigma$. If $H$ satisfies certain connectivity condition, then $\textrm{cr}_{\Sigma}(H\Box T)$ is not less than the sum of the crossing numbers of its ``subgraphs". In this paper, we introduced a new concept of generalized periodic graphs, which contains $H\Box C_n$. For a generalized periodic graph $G$ and a function $f(t)$, where $t$ is the number of subgraphs in a decomposition of $G$, we gave a necessary and sufficient condition for $\textrm{cr}_{\Sigma}(G)\geq f(t)$. As an application, we confirmed a conjecture of Lin et al. on the crossing number of the generalized Petersen graph $P(4h+2,2h)$ in the plane. Based on the condition, algorithms are constructed to compute lower bounds on the crossing number of generalized periodic graphs in $\Sigma$. In special cases, it is possible to determine lower bounds on an infinite family of generalized periodic graphs, by determining a lower bound on the crossing number of a finite generalized periodic graph.Comment: 26 pages, 20 figure

Arrangements, circular arrangements and the crossing number of C7×Cn

AbstractMotivated by the problem of determining the crossing number of the Cartesian product Cm×Cn of two cycles, we introduce the notion of an (m,n)-arrangement, which is a generalization of a planar drawing of Pn+1×Cm in which the two “end cycles” are in the same face of the remaining n cycles. The main result is that every (m,n)-arrangement has at least (m−2)n crossings. This is used to show that the crossing number of C7×Cn is 5n, in agreement with the general conjecture that the crossing number of Cm×Cn is (m−2)n, for 3⩽m⩽n