Convex Programming and Circumference of 3-Connected Graphs of Low Genus

The circumference of a graphGis the length of a longest cycle inG. In this paper, we shall show that, ifGis a 3-connected graph embeddable in the plane, the projective plane, the torus, or the Klein bottle, thenGhas circumference at least (1/6)×|V(G)|0.4+1. This improves a result of Jackson and Wormald.

[1]  E. L. Wilson,et al.  On Cycles and Connectivity in Planar Graphs , 1973, Canadian Mathematical Bulletin.

[2]  Zhicheng Gao,et al.  Spanning Planar Subgraphs of Graphs in the Torus and Klein Bottle , 1995, J. Comb. Theory, Ser. B.

[3]  W. T. Tutte.  A THEOREM ON PLANAR GRAPHS , 1956 .

[4]  Daniel P. Sanders.  On paths in planar graphs , 1997 .

[5]  D. Barnette.  Trees in Polyhedral Graphs , 1966, Canadian Journal of Mathematics.

[6]  J. Moon,et al.  Simple paths on polyhedra. , 1963 .

[7]  Edward A. Bender,et al.  Submaps of maps. I. General 0-1 laws , 1992, J. Comb. Theory, Ser. B.

[8]  Bill Jackson,et al.  Longest cycles in 3-connected planar graphs , 1992, J. Comb. Theory, Ser. B.

[9]  Nicholas C. Wormald,et al.  Random Triangulations of the Plane , 1988, Eur. J. Comb..

[10]  Hansjoachim Walther,et al.  Shortness Exponents of Families of Graphs , 1973, J. Comb. Theory, Ser. A.

[11]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .