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Cyclability, Connectivity and Circumference

Balachandran, N and Hebbar, A (2023) Cyclability, Connectivity and Circumference. In: 9th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2023, 9-11 February 2023, Gandhinagar, pp. 257-268.

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Official URL: https://doi.org/10.1007/978-3-031-25211-2_20


In a graph G, a subset of vertices S⊆ V(G) is said to be cyclable if there is a cycle containing the vertices in some order. G is said to be k-cyclable if any subset of k≥ 2 vertices is cyclable. If any k ordered vertices are present in a common cycle in that order, then the graph is said to be k-ordered. We show that when k≤n+3, k-cyclable graphs also have circumference c(G) ≥ 2 k, and that this is best possible. Furthermore when k≤3n4-1, c(G) ≥ k+ 2, and for k-ordered graphs we show c(G) ≥ min { n, 2 k}. We also generalize a result by Byer et al. [4] on the maximum number of edges in nonhamiltonian k-connected graphs, and show that if G is a k-connected graph of order n≥ 2 (k2+ k) with |E(G)|>(n-k2)+k2, then the graph is hamiltonian, and moreover the extremal graphs are unique.

Item Type: Conference Paper
Publication: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Publisher: Springer Science and Business Media Deutschland GmbH
Additional Information: The copyright for this article belongs to the Authors.
Keywords: Graph theory; Hamiltonians, Circumference; Connectivity; Cyclability; Cyclable; Extremal graph; Graph G; Hamiltonicity; K-connected graphs; Nonhamiltonians, Graphic methods
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 28 Mar 2023 09:59
Last Modified: 28 Mar 2023 09:59
URI: https://eprints.iisc.ac.in/id/eprint/81180

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