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arXiv · 2505.04131

The detour covering number and cummerbund covering number of a graph

Chengli Li·Xingzhi Zhan
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Abstract

We introduce several new concepts about graphs and investigate their basic properties. A longest path in a graph is called a detour and a longest cycle is called a cummerbund. The detour covering number of a graph is the number of vertices that lie in a detour. A graph is said to be detour covered if every vertex lies in a detour. The cummerbund covering number and cummerbund covered graphs are defined similarly. Some of the main results are as follows. (1) Minimum degree and forbidden subgraph conditions that ensure a graph to be cummerbund covered or detour covered. (2) The minimum cummerbund covering number and minimum detour covering number of a graph with connectivity or girth conditions. (3) The minimum cummerbund covering number of a $2$-connected bipartite graph and the extremal graphs.

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Chengli Li, Xingzhi Zhan. 2025-05-07. The detour covering number and cummerbund covering number of a graph. https://arxiv.org/abs/2505.04131

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