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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

## Abstract

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

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