Title The circumference of a graph with no K 3 , t-minor

It was shown by Chen and Yu that every 3-connected planar graph G contains a cycle of length at least |G|log3 , where |G| denotes the number of vertices of G. Thomas made a conjecture in a more general setting: there exists a function β(t) > 0 for t ≥ 3, such that every 3-connected graph G with no K3,t-minor, t ≥ 3, contains a cycle of length at least |G|β(t). We prove that this conjecture is true with β(t) = log8tt+1 2. We also show that every 2-connected graph with no K2,t-minor, t ≥ 3, contains a cycle of length at least |G|/tt−1.

[1]  Robin Thomas,et al.  Typical Subgraphs of 3- and 4-Connected Graphs , 1993, J. Comb. Theory, Ser. B.

[2]  Xingxing Yu,et al.  Disjoint paths, planarizing cycles, and spanning walks , 1997 .

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

[4]  Xingxing Yu,et al.  Long Cycles in 3-Connected Graphs , 2002, J. Comb. Theory, Ser. B.

[5]  Xingxing Yu,et al.  Convex Programming and Circumference of 3-Connected Graphs of Low Genus , 1997, J. Comb. Theory, Ser. B.

[6]  B. Grünbaum.  Polytopes, graphs, and complexes , 1970 .

[7]  Xingxing Yu,et al.  4-Connected Projective-Planar Graphs Are Hamiltonian , 1994, J. Comb. Theory, Ser. B.

[8]  Thomas Böhme,et al.  Ka, k Minors in Graphs of Bounded Tree-Width , 2002, J. Comb. Theory, Ser. B.

[9]  Thomas Böhme,et al.  Long Cycles in Graphs on a Fixed Surface , 2002, J. Comb. Theory, Ser. B.

[10]  Carsten Thomassen,et al.  A theorem on paths in planar graphs , 1983, J. Graph Theory.

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

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

[13]  H. Whitney.  A Theorem on Graphs , 1931 .

[14]  Brendan D. McKay,et al.  The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices , 1988, J. Comb. Theory, Ser. B.

[15]  Robin Thomas,et al.  Hamilton paths in toroidal graphs , 2005, J. Comb. Theory, Ser. B.

[16]  Xingxing Yu,et al.  Long cycles in 3-connected graphs in orientable surfaces , 2002, J. Graph Theory.