# The Crossing Numbers of Join of Paths and Cycles with Two Graphs of Order Five

• Marián Klešč
• Štefan Schrötter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7125)

## 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 are collected the exact values of crossing numbers for join of all graphs of at most four vertices with paths and cycles. In the paper, we extend these results. For two special graphs G on five vertices, we give the crossing numbers of the join products G + D n , G + P n , and G + C n , where D n consists on n isolated vertices, P n and C n are the path and cycle on n vertices, respectively.

## Keywords

graph drawing crossing number join product path cycle

## References

1. 1.
Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebraic and Discrete Methods 4, 312–316 (1983)
2. 2.
Kleitman, D.J.: The crossing number of K 5,n. J. Combin. Theory 9, 315–323 (1970)
3. 3.
Klešč, M.: The join of graphs and crossing numbers. Electronic Notes in Discrete Math. 28, 349–355 (2007)
4. 4.
Klešč, M.: The crossing numbers of join of the special graph on six vertices with path and cycle. Discrete Math. 310, 1475–1481 (2010)
5. 5.
Klešč, M., Schrötter, S.: The crossing numbers of join products of paths with graphs of order four. Discuss. Math. – Graph Theory 31, 321–331 (2011)
6. 6.
Kulli, V.R., Muddebihal, M.H.: Characterization of join graphs with crossing number zero. Far East J. Appl. Math. 5, 87–97 (2001)