Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SHARP ORE-TYPE CONDITIONS FOR THE EXISTENCE OF AN EVEN [4, b]-FACTOR IN A GRAPH

  • Cho, Eun-Kyung (Department of Mathematics Hankuk University of Foreign Studies) ;
  • Kwon, Su-Ah (Department of Applied Mathematics and Statistics The State University of New York) ;
  • O, Suil (Department of Applied Mathematics and Statistics The State University of New York)
  • Received : 2021.10.01
  • Accepted : 2022.01.14
  • Published : 2022.07.01
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Abstract

Let a and b be positive even integers. An even [a, b]-factor of a graph G is a spanning subgraph H such that for every vertex v ∈ V (G), dH(v) is even and a ≤ dH(v) ≤ b. Let κ(G) be the minimum size of a vertex set S such that G - S is disconnected or one vertex, and let σ2(G) = minuv∉E(G) (d(u)+d(v)). In 2005, Matsuda proved an Ore-type condition for an n-vertex graph satisfying certain properties to guarantee the existence of an even [2, b]-factor. In this paper, we prove that for an even positive integer b with b ≥ 6, if G is an n-vertex graph such that n ≥ b + 5, κ(G) ≥ 4, and σ2(G) ≥ ${\frac{8n}{b+4}}$, then G contains an even [4, b]-factor; each condition on n, κ(G), and σ2(G) is sharp.

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Acknowledgement

This work was financially supported by NRF 2020R1I1A1A0105858711. This work was financially supported by NRF 2020R1F1A1A01048226, NRF 2021K2A9A2A06044515, and NRF 2021K2A9A2A1110161711.

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