Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SHARP CONDITIONS FOR THE EXISTENCE OF AN EVEN [a, b]-FACTOR IN A GRAPH

  • Cho, Eun-Kyung (Department of Mathematics Hankuk University of Foreign Studies) ;
  • Hyun, Jong Yoon (Konkuk University) ;
  • O, Suil (Department of Applied Mathematics The State University of New York) ;
  • Park, Jeong Rye (Finance.Fishery.Manufacture Industrial Mathematics Center on Big Data Pusan National University)
  • Received : 2019.11.29
  • Accepted : 2020.10.16
  • Published : 2021.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let a and b be positive integers, and let V (G), ��(G), and ��2(G) be the vertex set of a graph G, the minimum degree of G, and the minimum degree sum of two non-adjacent vertices in V (G), respectively. 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, where dH(v) is the degree of v in H. Matsuda conjectured that if G is an n-vertex 2-edge-connected graph such that $n{\geq}2a+b+{\frac{a^2-3a}{b}}-2$, ��(G) ≥ a, and ${\sigma}_2(G){\geq}{\frac{2an}{a+b}}$, then G has an even [a, b]-factor. In this paper, we provide counterexamples, which are highly connected. Furthermore, we give sharp sufficient conditions for a graph to have an even [a, b]-factor. For even an, we conjecture a lower bound for the largest eigenvalue in an n-vertex graph to have an [a, b]-factor.

Keywords

References

  1. B. Bollobas, A. Saito, and N. C. Wormald, Regular factors of regular graphs, J. Graph Theory 9 (1985), no. 1, 97-103. https://doi.org/10.1002/jgt.3190090107
  2. Y. Egawa and H. Enomoto, Sufficient conditions for the existence of k-factors, in Recent studies in graph theory, 96-105, Vishwa, Gulbarga, 1989.
  3. X. Gu, Regular factors and eigenvalues of regular graphs, European J. Combin. 42 (2014), 15-25. https://doi.org/10.1016/j.ejc.2014.05.007
  4. T. Iida and T. Nishimura, An Ore-type condition for the existence of k-factors in graphs, Graphs Combin. 7 (1991), no. 4, 353-361. https://doi.org/10.1007/BF01787640
  5. P. Katerinis, Minimum degree of a graph and the existence of k-factors, Proc. Indian Acad. Sci. Math. Sci. 94 (1985), no. 2-3, 123-127. https://doi.org/10.1007/BF02880991
  6. S. Kim, S. O, J. Park, and H. Ree, An odd [1, b]-factor in regular graphs from eigenvalues, Discrete Math. 343 (2020), no. 8, 111906, 4 pp. https://doi.org/10.1016/j.disc.2020.111906
  7. M. Kouider and S. Ouatiki, Sufficient condition for the existence of an even [a, b]-factor in graph, Graphs Combin. 29 (2013), no. 4, 1051-1057. https://doi.org/10.1007/s00373-012-1168-9
  8. M. Kouider and P. D. Vestergaard, On even [2, b]-factors in graphs, Australas. J. Combin. 27 (2003), 139-147.
  9. M. Kouider and P. D. Vestergaard, Even [a, b]-factors in graphs, Discuss. Math. Graph Theory 24 (2004), no. 3, 431-441. https://doi.org/10.7151/dmgt.1242
  10. L. Lovasz, The factorization of graphs. II, Acta Math. Acad. Sci. Hungar. 23 (1972), 223-246. https://doi.org/10.1007/BF01889919
  11. H. Lu, Regular factors of regular graphs from eigenvalues, Electron. J. Combin. 17 (2010), p. #R159. https://doi.org/10.37236/431
  12. H. Lu, Regular graphs, eigenvalues and regular factors, J. Graph Theory 69 (2012), no. 4, 349-355. https://doi.org/10.1002/jgt.20581
  13. H. Lu, Z. Wu, and X. Yang, Eigenvalues and [1, n]-odd factors, Linear Algebra Appl. 433 (2010), no. 4, 750-757. https://doi.org/10.1016/j.laa.2010.04.002
  14. H. Matsuda, Ore-type conditions for the existence of even [2, b]-factors in graphs, Discrete Math. 304 (2005), no. 1-3, 51-61. https://doi.org/10.1016/j.disc.2005.09.009
  15. S. O, Eigenvalues and [a, b]-factors in regular graphs, Submitted.
  16. S. Tsuchiya and T. Yashima, A degree condition implying Ore-type condition for even [2, b]-factors in graphs, Discuss. Math. Graph Theory 37 (2017), no. 3, 797-809. https://doi.org/10.7151/dmgt.1964
  17. W. T. Tutte, The factors of graphs, Canad. J. Math. 4 (1952), 314-328. https://doi.org/10.4153/cjm-1952-028-2
  18. W. T. Tutte, Graph factors, Combinatorica 1 (1981), no. 1, 79-97. https://doi.org/10.1007/BF02579180
  19. O. Veblen, An application of modular equations in analysis situs, Ann. of Math. (2) 14 (1912/13), no. 1-4, 86-94. https://doi.org/10.2307/1967604