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The crossing numbers of join products of paths with three graphs of order five

Staš Michal, Švecová Mária

Opuscula Mathematica

2022

Vol. 42, no. 4

635--651

The main aim of this paper is to give the crossing number of the join product G∗ + Pn for the disconnected graph G∗ of order five consisting of the complete graph K4 and one isolated vertex, where Pn is the path on n vertices. The proofs are done with the help of a lot of well-known exact values for the crossing numbers of the join products of subgraphs of the graph G∗ with the paths. Finally, by adding new edges to the graph G∗, we are able to obtain the crossing numbers of the join products of two other graphs with the path Pn.

On the crossing numbers of join products of W4 + Pn and W4 + Cn

Stas Michal, Valiska Juraj

Opuscula Mathematica

2021

Vol. 41, no. 1

95-112

The crossing number cr(G) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main aim of the paper is to give the crossing number of the join product W4 + Pn and W4 + Cn for the wheel W4 on five vertices, where Pn and Cn are the path and the cycle on n vertices, respectively. Yue et al. conjectured that the crossing number of Wm + Cn is equal to [formula], for all m,n ≥ 3, and where the Zarankiewicz’s number[formula] is defined for n ≥ 1. Recently, this conjecture was proved for W3 + Cn by Klesc. We establish the validity of this conjecture for W4 + Cn and we also offer a new conjecture for the crossing number of the join product Wm + Pn for m ≥ 3 and n ≥ 2.

On the crossing numbers of join products of five graphs of order six with the discrete graph

Stas Michal

Opuscula Mathematica

2020

Vol. 40, no. 3

383--397

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G* + Dn, where the disconnected graph G* of order six consists of one isolated vertex and of one edge joining two nonadjacent vertices of the 5-cycle. In our proof, the idea of cyclic permutations and their combinatorial properties will be used. Finally, by adding new edges to the graph G*, the crossing numbers of Gi + Dn for four other graphs Gi of order six will be also established