Clique immersion in graph products

Collins, Karen L.; Heenehan, Megan E.; McDonald, Jessica

doi:10.48550/arxiv.1908.10457

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2021

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“…[3,4,14]). In [2] we define the immersion number of G, denoted im(G), to be the largest t for which G has a K t -immersion. Abu-Khzam and Langston [1] have conjectured that for a graph G, if the chromatic number of G is t, then im(G) ≥ t.…”

Section: Introductionmentioning

confidence: 99%

A Note on the Immersion Number of Generalized Mycielski Graphs

Collins¹,

Heenehan²,

McDonald³

2021

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The immersion number of a graph G, denoted im(G), is the largest t such that G has a Kt-immersion. In this note we are interested in determining the immersion number of the m-Mycielskian of G, denoted µm(G). Given the immersion number of G we provide a lower bound for im(µm(G)). To do this we introduce the "distinct neighbor property" of immersions. We also include examples of classes of graphs where im(µm(G)) exceeds the lower bound. We conclude with a conjecture about im(µm(Kt)).

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