On Dominating Even Subgraphs in Cubic Graphs
Pages 890 - 907
Abstract
It is known that a 3-edge-connected graph has a spanning even subgraph in which every component contains at least five vertices, and the lower bound is best possible. A natural question arises of whether we can improve the lower bound by changing the spanning property with the dominating property. In this paper, we show that a 3-edge-connected cubic graph has a dominating even subgraph in which every component contains at least six vertices.
References
[1]
N. Aggarwal, N. Garg, and S. Gupta, A 4/3-Approximation for TSP on Cubic 3-Edge-Connected Graphs, arXiv:1101.5586v1, 2011.
[2]
B. R. Alspach and C. D. Godsil, eds., Cycles in Graphs, Ann. Discrete Math. 27, North-Holland, Amsterdam, 1985.
[3]
P. Ash and B. Jackson, Dominating cycles in bipartite graphs, in Progress in Graph Theory, A. Bondy and U. S. R. Murty, eds., Academic Press, New York, 1984, pp. 81--87.
[4]
J. C. Bermond and C. Thomassen, Cycles in digraphs---a survey, J. Graph Theory, 5 (1981), pp. 1--43.
[5]
J. A. Bondy and U. S. R. Murty, Graph Theory, Grad. Texts in Math. 244, Springer, New York, 2008.
[6]
S. Boyd, R. Sitters, S. van der Ster, and L. Stougie, The traveling salesman problem on cubic and subcubic graphs, Math. Program., 144 (2014), pp. 227--245.
[7]
B. Candráková and R. Lukot'ka, Avoiding 5-circuits in 2-factors of cubic graphs, SIAM J. Discrete Math., 29 (2015), pp. 1387--1405.
[8]
B. Candráková and R. Lukot'ka, Cubic TSP---a 1.3-Approximation, arXiv:1506.06369v1, 2015.
[9]
J. Correa, O. Larré, and A. Soto, TSP tours in cubic graphs: Beyond 4/3, SIAM J. Discrete Math., 29 (2015), pp. 915--939.
[10]
H. Fleischner, Spanning Eulerian subgraphs, the splitting lemma, and Petersen's theorem, Discrete Math., 101 (1992), pp. 33--37.
[11]
H. Fleischner and B. Jackson, A note concerning some conjectures on cyclically 4-edge-connected 3-regular graphs, in Graph Theory in Memory of G. A. Dirac, L. D. Andersen, I. T. Jakobsen, C. Thomassen, B. Toft, P. D. Vestergaard, eds., Ann. Discrete Math. 41, North-Holland, Amsterdam, 1988, pp. 171--177.
[12]
D. Gamarnik, M. Lewenstein, and M. Sviridenko, An improved upper bound for the TSP in cubic 3-edge-connected graphs, Oper. Res. Lett., 33 (2005), pp. 467--474.
[13]
B. Jackson and K. Yoshimoto, Even subgraphs of bridgeless graphs and 2-factors of line graphs, Discrete Math., 307 (2007), pp. 2775--2785.
[14]
B. Jackson and K. Yoshimoto, Spanning even subgraphs of 3-edge-connected graphs, J. Graph Theory, 62 (2009), pp. 37--47.
[15]
T. Kaiser and R. Škrekovski, Cycles intersecting edge-cuts of prescribed sizes, SIAM J. Discrete Math., 22 (2008), pp. 861--874.
[16]
M. M. Matthews and D. P. Sumner, Hamiltonian results in $K_{1,3}$-free graphs, J. Graph Theory, 8 (1984), pp. 139--146.
[17]
J. Petersen, Die Theorie der regulären Graphs, Acta Math., 15 (1891), pp. 193--220.
[18]
Z. Ryjáček, On a closure concept in claw-free graphs, J. Combin. Theory. Ser. B, 70 (1997), pp. 217--224.
[19]
C. Thomassen, Reflections on graph theory, J. Graph Theory, 10 (1986), pp. 309--324.
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DOI:10.1137/sjdmec.31.2
Issue’s Table of Contents© 2017, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
United States
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Published: 01 January 2017
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