Which complete bipartite graphs Km, n have Hamilton cycles? Which have Hamilton paths?

Name: Problem 44, Chapter 11: Introductory Combinatorics
Uploaded: 2020-08-13T21:32:11-08:00
Duration: 1 min 24 s
Channel: Carson Merrill
Description: Which complete bipartite graphs Km, n have Hamilton cycles? Which have Hamilton paths?

step 1 So when we have a Hamilton circuit, this graph right here tells us that there is a set of M vert

Which complete bipartite graphs Km, n have Hamilton cycles?...

Key Concepts

Alternation Constraints in Bipartite Graphs

Due to the alternating nature of paths and cycles in bipartite graphs, there are specific constraints on the sizes of the two sets. For a Hamiltonian cycle to exist, the bipartite graph must have its two sets of equal size (and at least two vertices in each), whereas a Hamiltonian path can accommodate a slight imbalance (with one set having only one more vertex than the other) to maintain the required alternating sequence.

Complete Bipartite Graph

A complete bipartite graph, denoted by K?,?, is a graph whose vertices are divided into two disjoint sets of sizes m and n, with every vertex in one set connected to every vertex in the other. This structure inherently dictates that any cycle or path must alternate between contacts in the two sets, which directly affects conditions for the existence of Hamiltonian cycles and paths.

Hamiltonian Cycle

A Hamiltonian cycle is a cycle that visits every vertex of a graph exactly once and returns to the starting vertex. In bipartite graphs, such a cycle must alternate between the two vertex sets, which imposes the condition that the sets have the same size and contain at least two vertices. This balance is necessary to ensure that the route can alternate evenly between the two parts and eventually return to the start.

Hamiltonian Path

A Hamiltonian path is a path that visits every vertex of a graph exactly once but does not necessarily return to the starting point. In the context of bipartite graphs, the alternating requirement between the two parts means that the path can exist even if the two sets differ slightly in size—as long as the imbalance does not prevent the alternating sequence—typically requiring that the sizes differ by at most one.

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