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Anti-Ramsey numbers for disjoint copies of graphs

Gorgol I., Gorlich A.

Opuscula Mathematica

2017

Vol. 37, no. 4

567--575

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph G and a positive integer n, the anti-Ramsey number ar(n,G) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of H. Anti-Ramsey numbers were introduced by Erdos, Simonovits and Sós and studied in numerous papers. Let G be a graph with anti-Ramsey number ar(n, G). In this paper we show the lower bound for ar(n,pG), where pG denotes p vertex-disjoint copies of G. Moreover, we prove that in some special cases this bound is sharp.

Open trails in digraphs

Cichacz S., Gorlich A.

Opuscula Mathematica

2011

Vol. 31, no. 4

599-604

It has been shown in [S. Cichacz, A. Görlich, Decomposition of complete bipartite graphs into open trails, Preprint MD 022, (2006)] that any bipartite graph Ka,b, is decomposable into open trails of prescribed even lengths. In this article we consider the corresponding question for directed graphs. We show that the complete directed graphs ↔K n and ↔K a,b are arbitrarily decomposable into directed open trails.

Planar packing of cycles and unicyclic graphs

Gorlich A.

Demonstratio Mathematica

2009

Vol. 42, nr 4

673-679

We say that a graph G is packable into a complete graph Kn if there are two edge-disjoint subgraphs of Kn both isomorphic to G. It is equivalent to the existence of a permutation a of a vertex set in G such that if an edge xy belongs to E(G), then a(x)cr(y) does not belong to E(G). In 2002 Garcia et al. have shown that a non-star tree T is planary packable into a complete graph Kn. In this paper we show that for any packable cycle Cn except of the case n = 5 and n=7 there exists a planar packing into Kn. We also generalize this result to certain classes of unicyclic graphs.