The Hamiltonian problem and $t$-path traceable graphs

TL;DR

This paper explores the structure of maximal non-traceable and $t$-path traceable graphs, providing decomposition theorems and generalizing existing constructions to better understand their properties.

Contribution

It introduces a decomposition theorem for maximal $t$-path traceable graphs and extends Zelinka's construction to this broader class.

Findings

Decomposition reduces the problem to graphs with no universal vertex.

Traceability behavior analyzed under disjoint union and join operations.

Generalization of Zelinka's construction to $t$-path traceable graphs.

Abstract

The problem of characterizing maximal non-Hamiltonian graphs may be naturally extended to characterizing graphs that are maximal with respect to non-traceability and beyond that to $t$-path traceability. We show how traceability behaves with respect to disjoint union of graphs and the join with a complete graph. Our main result is a decomposition theorem that reduces the problem of characterizing maximal $t$-path traceable graphs to characterizing those that have no universal vertex. We generalize a construction of maximal non-traceable graphs by Zelinka to $t$-path traceable graphs.