Minimum crossings in join of graphs with paths and cycles

Abstract

The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. Only few results concerning crossing numbers of graphs obtained as join product of two graphs are known. There was collected the exact values of crossing numbers for join of all graphs of at most four vertices and of several graphs of order five with paths and cycles. We extend these results by giving the crossing numbers for join products of the special graph on six vertices with n isolated vertices as well as with the path on n vertices and with the cycle on n vertices.

Cyclic permutations in determining crossing numbers

“…Moreover, the exact values for crossing numbers of G + D n and G + P n for all graphs G of order at most four are given in [5]. The crossing numbers of the graphs G + P n and G + C n are also known for very few graphs G of order five and six; see [4,6], and [7]. In all these cases, the graph G is connected and contains at least one cycle.…”

“…The similar methods were partially used earlier in the papers [2,10]. We were unable to determine the crossing number of the join product G * + D n using the methods used in [4,6], and [7]. In Section 6 we refer the graph H on five vertices and six edges for which the crossing number of H + D n was obtained using previous methods; see also [6].…”

“…We know that, in D, at least three subgraphs T i do not cross G * . For these T i ∈ R 0 , we will discuss the existence of possible configurations of F i = G * ∪T i in the drawing D. Using the values in Table 2 we show that, in D, both conditions (4) and (5) of Lemma 5 hold, or both conditions (6) and (7) of Lemma 6 hold, or the condition (8) of Lemma 6 holds.…”

Determining Crossing Number of Join of the Discrete Graph with Two Symmetric Graphs of Order Five

“…In all of these cases, the graph G is connected and contains at least one cycle. Further, the exact values for the crossing numbers G + P n and G + C n have been also investigated for some graphs G of order five and six in [5,7,11,12].…”

On the Crossing Number of the Join of the Wheel on Five Vertices With the Discrete Graph

“…In all these cases, the graph G is usually connected and contains at least one cycle. The exact values of the crossing numbers G + P n and G + C n , where C n is the cycle with n vertices, have also been investigated for some graphs G of order five and six in [6,11,12,15,16,20,22].…”