Multithreshold multipartite graphs with small parts
Teeradej Kittipassorn, Thanaporn Sumalroj

TL;DR
This paper determines the exact threshold numbers for specific complete multipartite graphs with parts of size 3 and 4, advancing understanding of multithreshold graph classifications.
Contribution
It provides the exact threshold numbers for complete multipartite graphs with parts of size 3 and 4, resolving previously open cases.
Findings
Threshold number of $K_{3,3,...,3}$ and $K_{4,4,...,4}$ determined
Threshold numbers for their complements $nK_3$, $nK_4$ established
Improves upon previous results by Puleo
Abstract
A graph is a -threshold graph with thresholds if we can assign a real number to each vertex such that for any two distinct vertices and , is an edge if and only if the number of thresholds not exceeding is odd. The threshold number of a graph is the smallest for which it is a -threshold graph. Multithreshold graphs were introduced by Jamison and Sprague as a generalization of classical threshold graphs. They asked for the exact threshold numbers of complete multipartite graphs. Recently, Chen and Hao solved the problem for complete multipartite graphs where each part is not too small, and they asked for the case when each part has size . We determine the exact threshold numbers of , and their complements , . This improves a result of Puleo.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph theory and applications
