Let Ck denote a cycle of length k and let Sk denote a star with k edges. For multigraphs F, G and H, an (F,G)-decomposition of H is an edge decomposition of H into copies of F and G using at least one of each. For L⊆H and R⊆rH, an (F,G)-packing (resp. (F,G)-covering) of H with leave L (resp. padding R) is an (F,G)-decomposition of H-E(L) (resp. H+E(R)). An (F,G)-packing (resp. (F,G)-covering) of H with the largest (resp. smallest) cardinality is a maximum (F,G)-packing (resp. minimum (F,G)-covering), and its cardinality is referred to as the (F,G)-packing number (resp. (F,G)-covering number) of H. In this paper, we determine the packing number and the covering number of λKn,n with Ck's and Sk's for any λ, n and k, and give the complete solution of the maximum packing and the minimum covering of λKn,n with 4-cycles and 4-stars for any λ and n with all possible leaves and paddings.

Source : oai:HAL:hal-01188905v1

Volume: Vol. 16 no. 3 (in progress)

Section: Graph Theory

Published on: September 25, 2014

Submitted on: November 24, 2013

Keywords: complete bipartite multigraph,cycle,star,packing,covering,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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