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A note on Sullivan’s conjecture for oriented split graphs

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  • Zhao, Ruibo

Abstract

For a vertex v of an oriented graph, let d+(v) and d−(v) denote its out-degree and in-degree, respectively, and let d++(v) denote the number of vertices at distance exactly 2 from v. Sullivan (2006) conjectured that every oriented graph contains a vertex v with d++(v)≥d−(v). An oriented graph D is an oriented split graph if its vertex set can be partitioned into an independent set X and a set Y that induces a tournament. In this short note, we prove that Sullivan’s Conjecture holds for all source-free oriented split graphs with |X|=2 or |Y| ≤ 5 and source-free complete oriented split graphs with |X| ≤ 4.

Suggested Citation

  • Zhao, Ruibo, 2026. "A note on Sullivan’s conjecture for oriented split graphs," Applied Mathematics and Computation, Elsevier, vol. 524(C).
    • Handle: RePEc:eee:apmaco:v:524:y:2026:i:c:s0096300326000858
    DOI: 10.1016/j.amc.2026.130033
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