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On uniqueness of packing of three copies of 2-factors

Grzelec Igor, Madaras Tomáš, Onderko Alfréd

Opuscula Mathematica

2025

Vol. 45, no. 1

79--101

The packing of three copies of a graph G is the union of three edge-disjoint copies (with the same vertex set) of G. In this paper, we completely solve the problem of the uniqueness of packing of three copies of 2-regular graphs. In particular, we show that C3,C4,C5,C6 and 2C3 have no packing of three copies, C7,C8,C3∪C4,C4∪C4,C3∪C5 and 3C3 have unique packing, and any other collection of cycles has at least two distinct packings.

Local irregularity conjecture for 2-multigraphs versus cacti

Grzelec Igor, Woźniak Mariusz

Opuscula Mathematica

2024

Vol. 44, no. 1

49--65

A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph G such that every color induces a locally irregular submultigraph of G. A locally irregular colorable multigraph G is any multigraph which admits a locally irregular coloring. We denote by lir(G) the locally irregular chromatic index of a multigraph G, which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph G. In case of graphs the definitions are similar. The Local Irregularity Conjecture for 2-multigraphs claims that for every connected graph G, which is not isomorphic to K2, multigraph 2G obtained from G by doubling each edge satisfies lir(2G) ≤ 2. We show this conjecture for cacti. This class of graphs is important for the Local Irregularity Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims that every locally irregular colorable graph G satisfies lir(G) ≤ 3. At the beginning it has been observed that all not locally irregular colorable graphs are cacti. Recently it has been proved that there is only one cactus which requires 4 colors for a locally irregular coloring and therefore the Local Irregularity Conjecture was disproved.