Pancyclicity of Strong Products of Graphs

Hamiltonian threshold for strong products of graphs

“…This proves Conjecture 2 from [10,11]. Our proof is based on the concept of fractional factors in graphs which we introduce in Section 3.…”

“…We have managed to improve the bound h max ( ) ≤ ln + O( ) from [11] to h max ( ) ≤ (ln 2) + O( √ ). As noted in Introduction, the coefficient in the linear term is the best possible since the graph S (ln 2) is not hamiltonian.…”

“…In the final step of the proof, presented in Section 6, we find a 2-factor comprised by cycles of lengths at least and apply the following lemma which was implicitly proven in [11] (see the proof of Theorems 9 and 10 in [11]). Lemma 6.…”

“…In this section, we just introduce less standard concepts which we use and we also recall some results on factors in strong products of graphs mostly from [10,11].…”

Hamilton cycles in strong products of graphs

“…The following lower bound on hðS n Þ is due to Zaks [12]. We have conjectured [9] that the lower bound shown by Zaks is accurate:…”