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  1. ; ; (, Combinatorica)
    An analog of Dirac’s Theorem for Long Cycles in 2-Connected Graphs. 
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  2. ; ; ; ; (, SIAM Journal on Discrete Mathematics)
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  3. ; ; (, The Electronic Journal of Combinatorics)
    Let $D=(V,A)$ be a digraph. A vertex set $$K\subseteq V$$ is a quasi-kernel of $$D$$ if $$K$$ is an independent set in $$D$$ and for every vertex $$v\in V\setminus K$$, $$v$$ is at most distance 2 from $$K$$. In 1974, Chvátal and Lovász proved that every digraph has a quasi-kernel. P. L. Erdős and L. A. Székely in 1976 conjectured that if every vertex of $$D$$ has a positive indegree, then $$D$$ has a quasi-kernel of size at most $|V|/2$. This conjecture is only confirmed for narrow classes of digraphs, such as semicomplete multipartite, quasi-transitive, or locally semicomplete digraphs. In this note, we state a similar conjecture for all digraphs, show that the two conjectures are equivalent, and prove that both conjectures hold for a class of digraphs containing all orientations of 4-colorable graphs (in particular, of all planar graphs). 
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