On longest cycles in essentially 4-connected planar graphs

A planar 3-connected graph G is essentially 4-connected if, for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. Jackson and Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that . For a cubic essentially 4-connected planar graph G, Grünbaum with Malkevitch, and Zhang showed that G has a cycle on at least ¾ n vertices. In the present paper the result of Jackson and Wormald is improved. Moreover, new lower bounds on the length of a longest cycle of G are presented if G is an essentially 4-connected planar graph of maximum degree 4 or G is an essentially 4-connected maximal planar graph

Cubic graphs with large circumference deficit

The circumference $c(G)$ of a graph $G$ is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically $4$-, $5$- and $6$-edge-connected cubic graphs with circumference ratio $c(G)/|V(G)|$ bounded from above by $0.876$, $0.960$ and $0.990$, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically $4$-edge-connected cubic graph is at least $0.75$. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. H\"agglund and K. Markstr\"om, On stable cycles and cycle double covers of graphs with large circumference, Disc. Math. 312 (2012), 2540--2544]

EXTREMAL PROBLEMS CONCERNING CYCLES IN GRAPHS AND THEIR COMPLEMENTS

Let Gt(n) be the class of connected graphs on n vertices having the longest cycle of length t and let G ∈ Gt(n). Woodall (1976) determined the maximum number of edges of G, ε(G) ≤ w(n,t), where w(n, t) = (n - 1) t/2 - r(t – r - 1)/2 and r = (n - 1 ) - (t - 1) ⎣(n - 1)/(t - 1)⎦. An alternative proof and characterization of the extremal (edge-maximal) graphs given by Caccetta and Vijayan (1991). The edge- maximal graphs have the property that their complements are either disconnected or have a cycle going through each vertex (i.e. they are hamiltonian). This motivates us to investigate connected graphs with prescribed circumference (length of the longest cycle) having connected complements with cycles . More specifically, we focus our investigations on : Let G(n, c, c ) denote the class of connected graphs on n vertices having circumference c and whose connected complements have circumference c . The problem of interest is that of determining the bounds of the number of edges of a graph G ∈ G(n, c, c ) and characterize the extremal graphs of G(n, c, c ). We discuss the class G(n, c, c ) and present some results for small c. In particular for c = 4 and c = n - 2, we provide a complete solution. Key words : extremal graph, circumferenc

On The Graphs and Their Complements with Prescribed Circumference

Let Gt(n) be the class of connected graphs on n vertices having the longest cycle of length t and let G ∈ Gt(n). Woodall (1976) determined the maximum number of edges of G. An alternative proof and characterization of the extremal (edge-maximal) graphs given by Caccetta & Vijayan (1991). The edge-maximal graphs have the property that their complements are either disconnected or have a cycle going through each vertex (i.e. they are hamiltonian). This motivates us to investigate connected graphs with prescribed circumference (length of the longest cycle) having connected complements with cycles . More specifically, we focus our investigations on the class G (n, c, c) denoting the class of connected graphs on n vertices having circumference c and whose connected complements have circumference c. The problem of interest is that of determining the bounds of the number of edges of a graph G∈ G(n, c, c) and characterize the extremal graphs of G(n, c, c). We discuss the class G (n, c, c) and present some results for small c. In particular for c=4 and c =n-2, we provide a complete solution

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 $G$ 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 $c(G) \geq n - 3$ or $c^*(G) \geq m - n - 1$. 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 $P(n, k)$, $1 \leq k < \frac{n}{2}$, the co-spectrum of $P(n, k)$ is $\{3, 4, 5, ..., n+2\}$.Comment: 16 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.pd