Given graphs G and H , an H -decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H . Let φ H (n) be the smallest number φ such that any graph G of order n admits an H -decomposition with at most φ parts.Here we determine the asymptotic of φ H (n) for any fixed graph H as n tends to infinity. The exact computation of φ H (n) for an arbitrary H is still an open problem. Bollobás [B. Bollobás, On complete subgraphs of different orders, Math. Proc. Cambridge Philos. Soc. 79 (1976) 19-24] accomplished this task for cliques. When H is bipartite, we determine φ H (n) with a constant additive error and provide an algorithm returning the exact value with running time polynomial in log n.
Given two graphs G and H, an H‐decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. Let φ(n,H) be the smallest number ϕ such that any graph G of order n admits an H‐decomposition with at most ϕ parts. Pikhurko and Sousa conjectured that φ(n,H)= ex (n,H) for χ(H)≥3 and all sufficiently large n, where ex (n,H) denotes the maximum number of edges in a graph on n vertices not containing H as a subgraph. Their conjecture has been verified by Özkahya and Person for all edge‐critical graphs H. In this article, the conjecture is verified for the k‐fan graph. The k‐fan graph, denoted by Fk, is the graph on 2k+1 vertices consisting of k triangles that intersect in exactly one common vertex called the center of the k‐fan.
In this paper we consider the problem of finding the smallest number $q$ such that any graph $G$ of order $n$ admits a decomposition into edge disjoint copies of a fixed graph $H$ and single edges with at most $q$ elements. We solve the case when $H$ is the 5-cycle, the 5-cycle with a chord and any connected non-bipartite non-complete graph of order 4.
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