Intersection of Longest Cycle and Largest Bond in 3-Connected Graphs · arXivDesk
2305.15110May 22, 202316 pages, 19 figures. Paper presented at the 54th Southeastern International Conference on Combinatorics, Graph Theory and Computing (March 6-10, 2023); submitted on May 9, 2023 to the conference proceedings book series publication titled "Springer Proceedings in Mathematics and Statistics" (PROMS). Paper abstract also on https://www.math.fau.edu/combinatorics/abstracts/ren54.pdf
Intersection of Longest Cycle and Largest Bond in 3-Connected Graphs
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
Nearby in the stack
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