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arXiv · 2010.10790

The Second Neighborhood Conjecture for Oriented Graphs Missing $\{C_{4}, \overline{C_{4}}, S_{3},$ chair and co-chair$\}$-Free Graph

Darine Al Mniny·Salman Ghazal
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Abstract

Seymour's Second Neighborhood Conjecture (SNC) asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. In this paper, we prove that if $G$ is a graph containing no induced $C_4$, $\overline{C_4}$, $S_3$, chair and $\overline{chair}$, then every oriented graph missing $G$ satisfies this conjecture. As a consequence, we deduce that the conjecture holds for every oriented graph missing a threshold graph, a generalized comb or a star.

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Darine Al Mniny, Salman Ghazal. 2020-10-21. The Second Neighborhood Conjecture for Oriented Graphs Missing $\{C_{4}, \overline{C_{4}}, S_{3},$ chair and co-chair$\}$-Free Graph. https://arxiv.org/abs/2010.10790

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