Abstract
Given a connected graphG, we say that a setC ⊆V(G) is convex inG if, for every pair of verticesx, y ∈ C, the vertex set of everyx-y geodesic inG is contained inC. The convexity number ofG is the cardinality of a maximal proper convex set inG. In this paper, we show that every pairk, n of integers with 2 ≤k ≤ n−1 is realizable as the convexity number and order, respectively, of some connected triangle-free graph, and give a lower bound for the convexity number ofk-regular graphs of ordern withn>k+1.
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Byung Kee Kim received his Ph. D from Korea University. He is now a professor of Chongju University. His research interest focuses on convexity in graphs and topology.
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Kim, B.K. A lower bound for the convexity number of some graphs. JAMC 14, 185–191 (2004). https://doi.org/10.1007/BF02936107
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DOI: https://doi.org/10.1007/BF02936107