
TL;DR
This paper identifies and classifies the only two 4-critical t-perfect graphs within the class of complements of line graphs, advancing understanding of chromatic bounds in t-perfect graph theory.
Contribution
It introduces a new example of a 4-critical t-perfect graph and proves these are the only such graphs in the class of complements of line graphs, also extending results to h-perfect graphs.
Findings
Identified the complement of the line graph of W_5 as a 4-critical t-perfect graph.
Proved these are the only 4-critical t-perfect graphs in the class of complements of line graphs.
Showed that all t-perfect P_6-free graphs are 4-colorable.
Abstract
It is an open question whether the chromatic number of -perfect graphs is bounded by a constant. The largest known value for this parameter is 4, and the only example of a 4-critical -perfect graph, due to Laurent and Seymour, is the complement of the line graph of the prism (a graph is 4-critical if it has chromatic number 4 and all its proper induced subgraphs are 3-colorable). In this paper, we show a new example of a 4-critical -perfect graph: the complement of the line graph of the 5-wheel . Furthermore, we prove that these two examples are in fact the only 4-critical -perfect graphs in the class of complements of line graphs. As a byproduct, an analogous and more general result is obtained for -perfect graphs in this class. The class of -free graphs is a proper superclass of complements of line graphs and appears as a natural candidate to further…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Interconnection Networks and Systems
