Hamiltonian Cycles in the Square of a Graph

Abstract

We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of the square of a connected graph such that every vertex of degree at least three in a block graph corresponds to a cut vertex and any two these vertices are at distance at least four.

Revisiting the Hamiltonian theme in the square of a block: the general case

“…In subsequent papers we shall use some of the theorems of this paper to describe (among other results) the most general structure a graph may have such that its square is hamiltonian or hamiltonian connected, respectivelly. This will also solve a problem raised in [4] in the affirmative and proves a conjecture raised in [19]; we shall also present a partial solution of a conjecture stated in [5].…”

Connected Even Factors in the Square of Essentially 2-Edge-connected Graph

“…Then G ′ is a connected graph without non-trivial bridges and without bad leaves and we prove by induction on k that (G ′ ) 2 contains a [2,4]-factor F ′ such that 1) d F ′ (x) = 2 for any vertex x that is not a cut vertex of G;…”

“…If G is a connected graph such that every induced S(K 1,3 ) has at least three edges in a block of degree at most 2, then G 2 is hamiltonian. Theorem 2 was generalized by Ekstein et al in [2] for [2, 2s]-factors.…”

“…By Theorem 1, there is a Hamiltonian cycle C in (B k ) 2 such that two edges f 1 , f 2 of C incident with v 0 belong to B k and thus belong to G ′ . [2,4]-factor of (G ′ ) 2 with properties 1), 2), and 3). Hence v 0 belongs to exactly two blocks of G ′ and F ′ = F 2 ∪ C is the [2, 4]-factor of (G ′ ) 2 with properties 1), 2), and 3).…”

“…There are several papers (e.g. see [2], [4], [5], [6], [7], [8], [9], and [10]) about hamiltonian properties in the square of a graph. This paper deals with connected even factors which generalize some previous known results.…”