The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. Recently, the crossing numbers of join products of two graphs have been studied. In the paper, we extend know results concerning crossing numbers of join products of small graphs with discrete graphs. The crossing number of the join product G * + D n for the disconnected graph G * consisting of five vertices and of three edges incident with the same vertex is given. Up to now, the crossing numbers of G + D n were done only for connected graphs G. In the paper also the crossing numbers of G * + P n and G * + C n are given. The paper concludes by giving the crossing numbers of the graphs H + D n , H + P n , and H + C n for four different graphs H with |E(H)| ≤ |V (H)|. The methods used in the paper are new. They are based on combinatorial properties of cyclic permutations.
In this paper, we show the values of crossing numbers for join products of graph G on five vertices with the discrete graph D n and the path P n on n vertices. The proof is done with the help of software. The software generates all cyclic permutations for a given number n. For cyclic permutations, P 1 -P m will create a graph in which to calculate the distances between all vertices of the graph. These distances are used in proof of crossing numbers of presented graphs.
The crossing number cr(G) of a graph G is the minimal number of edge crossings over all drawings of G in the plane. In the paper, we extend results of the exact values of crossing numbers for join of graphs of order five. We give the crossing number of the join product G + D n , where the graph G consists of one 4-cycle and one isolated vertex, and D n consists on n isolated vertices.
The crossing number \(\mathrm{cr}(G)\) 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 \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\).
The main aim of the paper is to give the crossing number of join product G + D n for the connected graph G of order five isomorphic with the complete tripartite graph K 1,1,3 , where D n consists on n isolated vertices. The proof of the crossing number of K 1,1,3,n was published by very rather unclear discussion of cases by Ho in [5]. In our proofs, it will be extend the idea of the minimum numbers of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph G onto the set of subgraphs which cross the edges of the graph G exactly once. The methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations. Finally, by adding one edge to the graph G, we are able to obtain the crossing number of the join product with the discrete graph D n for one new graph.
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