Intersection of Longest Cycle and Largest Bond in 3-Connected Graphs
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A bond in a graph is a minimal nonempty edge-cut. A connected graph G is dual Hamiltonian if the vertex set can be partitioned into two subsets X and Y such that the subgraphs induced by X and Y are both trees. There is much interest in studying the longest cycles and largest bonds in graphs. H. Wu conjectured that any longest cycle must meet any largest bond in a simple 3-connected graph. In this paper, the author proves that the above conjecture is true for certain classes of 3-connected graphs: Let G be a simple 3-connected graph with n vertices and m edges. Suppose c(G) is the size of a longest cycle, and c^*(G) is the size of a largest bond. Then each longest cycle meets each largest bond if either c(G) ≥ n - 3 or c^*(G) ≥ m - n - 1. Sanford determined in her Ph.D. thesis the cycle spectrum of the well-known generalized Petersen graph P(n, 2) (n is odd) and P(n, 3) (n is even). Flynn proved in her honors thesis that any generalized Petersen graph P(n, k) is dual Hamiltonian. The author studies the bond spectrum (called the co-spectrum) of the generalized Petersen graphs and extends Flynn's result by proving that in any generalized Petersen graph P(n, k), 1 ≤ k < n/2, the co-spectrum of P(n, k) is {3, 4, 5, ..., n+2}.
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