We investigate the orientable genus of Gn, the cartesian product of n triangles, with a particular attention paid to the two smallest unsolved cases n = 4 and 5. Using a lifting method we present a general construction of a low-genus embedding of Gn using a low-genus embedding of Gn−1. Combining this method with a computer search and a careful analysis of face structure we show that 30 6 γ(G4) ...

We examine the hamiltonicity of the cartesian product P = G1 ×G2 of two graphs G1, G2. We provide necessary and/or sufficient conditions for P to be hamiltonian, depending on the hamiltonian properties of G1 and G2, with corresponding constructions. We also prove a conjecture by Batagelj and Pisanski related to the ‘cyclic hamiltonicity’ of a graph.

9 We study the inheritance of path-pairability in the Cartesian product of 10 graphs and prove additive and multiplicative inheritance patterns of path11 pairability, depending on the number of vertices in the Cartesian product. 12 We present path-pairable graph families that improve the known upper 13 bound on the minimal maximum degree of a path-pairable graph. Further 14 results and open que...

Cartesian products of graphs have been studied extensively since the 1960s. They make it possible to decrease the algorithmic complexity of problems by using the factorization of the product. Hypergraphs were introduced as a generalization of graphs and the definition of Cartesian products extends naturally to them. In this paper, we give new properties and algorithms concerning coloring aspect...

For a connected graph G = (V (G), E(G)), a vertex set S ⊆ V (G) is a k-restricted vertex-cut if G − S is disconnected such that every component of G − S has at least k vertices. The k-restricted connectivity κk(G) of the graph G is the cardinality of a minimum k-restricted vertex-cut of G. In this paper, we give the 3-restricted connectivity and the 4-restricted connectivity of the Cartesian pr...

The matching preclusion number of a graph G, mp(G), is the minimum number of edges whose deletion leaves a resulting graph that has neither perfect matchings nor almost perfect matchings. Besides its theoretical linkage with conditional connectivity and extremal graph theory, the matching preclusion number is a measure of robustness in interconnection networks. In this paper we develop general ...

Given a Cartesian product G = GI x x G,, (m > 2) of nontrivial connected graphs G, and the base d, dimension D de Bruijn graph B(d,D), it is investigated under which conditions G is (or is not) a subgraph of B(d, D). We present a complete solution of this problem for the case D 3 4. For D = 3, we give partial results including a complete solution for the case that G is a torus, i.e., G is the C...

The purpose of this paper is to present some coupled fixed point results on a metric space endowed with two $b$-metrics. We shall apply a fixed point theorem for an appropriate operator on the Cartesian product of the given spaces endowed with directed graphs. Data dependence, well-posedness and Ulam-Hyers stability are also studied. The results obtained here will be applied to prove the existe...

The distance $d(u,v)$ between two vertices $u$ and $v$ of a graph $G$ is equal to the length of a shortest path that connects $u$ and $v$. Define $WW(G,x) = 1/2sum_{{ a,b } subseteq V(G)}x^{d(a,b) + d^2(a,b)}$, where $d(G)$ is the greatest distance between any two vertices. In this paper the hyper-Wiener polynomials of the Cartesian product, composition, join and disjunction of graphs are compu...