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On sufficient conditions for Hamiltonicity of graphs, and beyond

Author

Listed:
  • Hechao Liu

    (Hubei Normal University)

  • Lihua You

    (South China Normal University)

  • Yufei Huang

    (Guangzhou Civil Aviation College)

  • Zenan Du

    (South China Normal University)

Abstract

Identifying certain conditions that ensure the Hamiltonicity of graphs is highly important and valuable due to the fact that determining whether a graph is Hamiltonian is an NP-complete problem.For a graph G with vertex set V(G) and edge set E(G), the first Zagreb index ( $$M_{1}$$ M 1 ) and second Zagreb index ( $$M_{2}$$ M 2 ) are defined as $$M_{1}(G)=\sum \limits _{v_{i}v_{j}\in E(G)}(d_{G}(v_{i})+d_{G}(v_{j}))$$ M 1 ( G ) = ∑ v i v j ∈ E ( G ) ( d G ( v i ) + d G ( v j ) ) and $$M_{2}(G)=\sum \limits _{v_{i}v_{j}\in E(G)}d_{G}(v_{i})d_{G}(v_{j})$$ M 2 ( G ) = ∑ v i v j ∈ E ( G ) d G ( v i ) d G ( v j ) , where $$d_{G}(v_{i})$$ d G ( v i ) denotes the degree of vertex $$v_{i}\in V(G)$$ v i ∈ V ( G ) . The difference of Zagreb indices ( $$\Delta M$$ Δ M ) of G is defined as $$\Delta M(G)=M_{2}(G)-M_{1}(G)$$ Δ M ( G ) = M 2 ( G ) - M 1 ( G ) .In this paper, we try to look for the relationship between structural graph theory and chemical graph theory. We obtain some sufficient conditions, with regards to $$\Delta M(G)$$ Δ M ( G ) , for graphs to be k-hamiltonian, traceable, k-edge-hamiltonian, k-connected, Hamilton-connected or k-path-coverable.

Suggested Citation

  • Hechao Liu & Lihua You & Yufei Huang & Zenan Du, 2024. "On sufficient conditions for Hamiltonicity of graphs, and beyond," Journal of Combinatorial Optimization, Springer, vol. 47(2), pages 1-12, March.
  • Handle: RePEc:spr:jcomop:v:47:y:2024:i:2:d:10.1007_s10878-024-01110-4
    DOI: 10.1007/s10878-024-01110-4
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    References listed on IDEAS

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    1. Wang, Yiqiao & Zheng, Lina, 2020. "Computation on the difference of Zagreb indices of maximal planar graphs with diameter two," Applied Mathematics and Computation, Elsevier, vol. 377(C).
    2. Shuchao Li & Licheng Zhang & Minjie Zhang, 2019. "On the extremal cacti of given parameters with respect to the difference of zagreb indices," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 421-442, August.
    3. Yong Lu & Qiannan Zhou, 2021. "On sufficient topological indices conditions for properties of graphs," Journal of Combinatorial Optimization, Springer, vol. 41(2), pages 487-503, February.
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