The detour covering number and cummerbund covering number of a graph
Chengli Li, Xingzhi Zhan
TL;DR
The paper investigates how long detours (longest paths) and cummerbunds (longest cycles) cover a graph by introducing the detour covering number $\mathrm{dc}(G)$ and the cummerbund covering number $\mathrm{cc}(G)$ and studying when graphs are detour-covered or cummerbund-covered under various structural conditions. It provides tight extremal and constructive results, including degree and connectivity thresholds, forbidden-subgraph characterizations, and bipartite/girth-specific bounds, with explicit extremal graphs and sharp minima (e.g., $\mathrm{cc}(G)$ and $\mathrm{dc}(G)$ for $k$-connected graphs). Key outcomes include separation results (Theorem 1), threshold-degree results (Theorems 3–5), exact minima for connected and $k$-connected graphs (Theorems 7–8), induced-subgraph and threshold graph consequences (Theorems 10–14), and girth-based bounds (Theorem 16). The work lays a framework for understanding coverage by long cycles and paths and proposes several open problems for further exploration.
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.