20.500.12556/RUL-136571 Remarks on the local irregularity conjecture A locally irregular graph is a graph in which the end vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it allows a locally irregular edge coloring. The locally irregular chromatic index of a colorable graph G, denoted by χ$^′_{irr}$(G), is the smallest number of colors used by a locally irregular edge coloring of G. The local irregularity conjecture claims that all graphs, except odd-length paths, odd-length cycles and a certain class of cacti are colorable by three colors. As the conjecture is valid for graphs with a large minimum degree and all non-colorable graphs are vertex disjoint cacti, we study rather sparse graphs. In this paper, we give a cactus graph B which contradicts this conjecture, i.e., χ$^′_{irr}$(B) = 4. Nevertheless, we show that the conjecture holds for unicyclic graphs and cacti with vertex disjoint cycles. locally irregular edge coloring local irregularity conjecture unicyclic graph cactus graph lokalno iregularno barvanje povezav domneva o lokalni iregularnosti uniciklični graf kaktus graf true false true Angleški jezik Slovenski jezik Članek v reviji 2022-05-11 14:54:07 2022-05-11 14:54:08 2022-09-15 04:14:01 0000-00-00 00:00:00 2021 0 0 10 str. iss. 24, art. 3209 Vol. 9 2021 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 2227-7390 10.3390/math9243209 93453315 mathematics-09-03209-v2.pdf mathematics-09-03209-v2.pdf 1 7595A5813D63796FEF468EA06ABE17DA 5f7dc6ec4cb8cce99f1e355690b32424e5dc26eaedd187f29c55962088560f0a f3aaae24-d129-11ec-8aca-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=156278 https://www.mdpi.com/2227-7390/9/24/3209 1 5740f459-d129-11ec-8aca-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=156277 Fakulteta za matematiko in fiziko 0 0 0