On plane cycles in geometric multipartite graphs

TL;DR

This paper investigates the existence and properties of plane cycles in geometric complete multipartite graphs, providing new theorems, characterizations, algorithms, and complexity results related to plane Hamiltonian cycles.

Contribution

It introduces new bounds on plane cycle lengths, characterizes certain plane cycles, and develops algorithms and complexity results for plane Hamiltonian cycles in geometric bipartite graphs.

Findings

A plane cycle of length t ≥ 6 implies a smaller cycle of length at least ⌊t/2⌋ + 1.

Characterization of geometric complete multipartite graphs with plane cycles involving repeated color classes.

An algorithm with complexity O(n log n + nk^2) + O(k^{5k}) for detecting plane Hamiltonian cycles in K_{n,n}.

Abstract

A geometric graph is a drawing of a graph in the plane where the vertices are drawn as points in general position and the edges as straight-line segments connecting their endpoints. It is plane if it contains no crossing edges. We study plane cycles in geometric complete multipartite graphs. We prove that if a geometric complete multipartite graph contains a plane cycle of length $t$, with $t \geq 6$, it also contains a smaller plane cycle of length at least $\lfloor t/2\rfloor + 1$. We further give a characterization of geometric complete multipartite graphs that contain plane cycles with a color class appearing at least twice. For geometric drawings of $K_{n,n}$, we give a sufficient condition under which they have, for each $s \leq n$, a plane cycle of length 2s. We also provide an algorithm to decide whether a given geometric drawing of $K_{n,n}$ contains a plane Hamiltonian cycle in time $O(n \log n + nk^2) + O(k^{5k})$, where k is the number of vertices inside the convex hull of all vertices. Finally, we prove that it is NP-complete to decide if a subset of edges of a geometric complete bipartite graph H is contained in a plane Hamiltonian cycle in H.