Abstract
A graph with n vertices is said to have a small cycle cover provided its edges can be covered with at most (2n − 1)/3 cycles. Bondy [2] has conjectured that every 2-connected graph has a small cycle cover. In [3] Lai and Lai prove Bondy’s conjecture for plane triangulations. In [1] the author extends this result to all planar 3-connected graphs, by proving that they can be covered by at most (n + 1)/2 cycles. In this paper we show that Bondy’s conjecture holds for all planar 2-connected graphs. We also show that all planar 2-edge-connected graphs can be covered by at most (3n − 3)/4 cycles and we show an infinite family of graphs for which this bound is attained.
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References
Barnette, D.W.: Cycle covers of planar 3-connected graphs. Journal of Combinatorial Mathematics and Combinatorial Computing (to be published)
Bondy, J.A.: Small cycle double covers of graphs, Cycles and Rays. pp. 21–40. Dordecht: Kluwer Academic 1990
Lai, H.-J., Lai, H.: Cycle covering of plane triangulations. Journal of Combinatorial Mathematics and Combinatorial Computing 10, 3–12 (1991)
Lai, H.-J.: Small cycle covers of planar graphs. Congress Numerantium (Proceedings for 1995 Southeastern Conference on Combinatorics, Graph Theory and Computing) (to be published)
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Barnette, D.W. Cycle Covers of Planar 2-Edge-Connected Graphs. Graphs and Combinatorics 13, 315–323 (1997). https://doi.org/10.1007/BF03353010
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DOI: https://doi.org/10.1007/BF03353010