# BIPACKING A BIPARTITE GRAPH WITH GIRTH AT LEAST 12

Wang, Hong

• Received : 2018.01.02
• Accepted : 2018.09.19
• Published : 2019.01.01
• 188 7

#### Abstract

Let G be a bipartite graph with (X, Y ) as its bipartition. Let B be a complete bipartite graph with a bipartition ($V_1$, $V_2$) such that $X{\subseteq}V_1$ and $Y{\subseteq}V_2$. A bi-packing of G in B is an injection ${\sigma}:V(G){\rightarrow}V(B)$ such that ${\sigma}(X){\subseteq}V_1$, ${\sigma}(Y){\subseteq}V_2$ and $E(G){\cap}E({\sigma}(G))={\emptyset}$. In this paper, we show that if G is a bipartite graph of order n with girth at least 12, then there is a complete bipartite graph B of order n + 1 such that there is a bi-packing of G in B. We conjecture that the same conclusion holds if the girth of G is at least 8.

#### Keywords

packing;embedding;placement

#### References

1. B. Bollobas, Extremal Graph Theory, London Mathematical Society Monographs, 11, Academic Press, Inc., London, 1978.
2. S. Brandt, Embedding graphs without short cycles in their complements, in Graph theory, combinatorics, and algorithms, Vol. 1, 2 (Kalamazoo, MI, 1992), 115-121, Wiley-Intersci. Publ, Wiley, New York, 1995.
3. R. J. Faudree, C. C. Rousseau, R. H. Schelp, and S. Schuster, Embedding graphs in their complements, Czechoslovak Math. J. 31(106) (1981), no. 1, 53-62.
4. J.-L. Fouquet and A. P. Wojda, Mutual placement of bipartite graphs, Discrete Math. 121 (1993), no. 1-3, 85-92. https://doi.org/10.1016/0012-365X(93)90540-A
5. A. Gorlich, M. Polisniak, M.Wozniak, and A. Ziolo, A note on embedding graphs without short cycles, Discrete Math. 286 (2004), no. 1-2, 75-77. https://doi.org/10.1016/j.disc.2003.11.048
6. B. Orchel, 2-placement of (p, q)-trees, Discuss. Math. Graph Theory 23 (2003), no. 1, 23-36. https://doi.org/10.7151/dmgt.1183
7. N. Sauer and H. Wang, The chromatic number of the two packings of a forest, Math. Yellow Series of University of Calgary (1992), No. 732.
8. H. Wang, Packing two forests into a bipartite graph, J. Graph Theory 23 (1996), no. 2, 209-213. https://doi.org/10.1002/(SICI)1097-0118(199610)23:2<209::AID-JGT12>3.0.CO;2-B
9. H. Wang, Packing two bipartite graphs into a complete bipartite graph, J. Graph Theory 26 (1997), no. 2, 95-104. https://doi.org/10.1002/(SICI)1097-0118(199710)26:2<95::AID-JGT4>3.0.CO;2-A