Locally Irregular Total Colorings of Graphs
Anna Flaszczy\'nska, Aleksandra Gorzkowska, Igor Grzelec, Alfr\'ed Onderko, Mariusz Wo\'zniak

TL;DR
This paper investigates locally irregular total colorings of graphs, proving the conjecture for specific graph classes and establishing bounds based on graph properties like chromatic number, planarity, and graph decompositions.
Contribution
It proves the conjecture that two colors suffice for locally irregular total coloring in certain graph classes and provides bounds related to graph chromatic number and planarity.
Findings
Proved the conjecture for cacti, subcubic, and split graphs.
Established bounds based on chromatic number.
Provided constant bounds for planar and outerplanar graphs.
Abstract
A total graph is an ordered triple , where are the sets of empty and full vertices, respectively, , and the set of edges is a subset of \(\binom{V_0 \cup V_1}{2}\) . A simple graph is a total graph in which all vertices are full. We say that a total graph is locally irregular if every two adjacent vertices have different total degrees, where by the total degree of a vertex in we mean the number of edges in that contain plus 1 if is full, or plus 0 if is empty. A total coloring of a graph whose colors induce locally irregular total subgraphs is called locally irregular total coloring, and the minimum number of colors required in such a coloring of is denoted by . In 2015, Baudon, Bensmail, Przyby{\l}o, and Wo\'zniak conjectured that ${\rm…
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Advanced Graph Theory Research · Limits and Structures in Graph Theory
