## Abstract

A graph *G*=(*V*,*E*) is called a split graph if there exists a partition *V*=*I*∪*K* such that the subgraphs *G*[*I*] and *G*[*K*] of *G* induced by *I* and *K* are empty and complete graphs, respectively. Burkard and Hammer gave a necessary condition for a split graph *G* with |*I*|<|*K*| to be Hamiltonian (J. Comb. Theory, Ser. B 28:245–248, 1980). We will call a split graph *G* with |*I*|<|*K*| satisfying this condition a Burkard–Hammer graph. Further, a split graph *G* is called a maximal nonhamiltonian split graph if *G* is nonhamiltonian but *G*+*uv* is Hamiltonian for every \(uv\not\in E\), where *u*∈*I* and *v*∈*K*. N.D. Tan and L.X. Hung have classified maximal nonhamiltonian Burkard–Hammer graphs *G* with minimum degree *δ*(*G*)≥|*I*|−3. Recently, N.D. Tan and Iamjaroen have classified maximal nonhamiltonian Burkard–Hammer graphs with |*I*|≠6,7 and *δ*(*G*)=|*I*|−4. In this paper, we complete the classification of maximal nonhamiltonian Burkard–Hammer graphs with *δ*(*G*)=|*I*|−4 by finding all such graphs for the case |*I*|=6,7.

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## Acknowledgements

I would like to express my sincere thanks to the referee for his valuable remarks which helped me to improve the paper.

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Tan, N.D. The Completion of a Classification for Maximal Nonhamiltonian Burkard–Hammer Graphs.
*Viet J Math* **41**, 465–505 (2013). https://doi.org/10.1007/s10013-013-0046-y

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DOI: https://doi.org/10.1007/s10013-013-0046-y

### Keywords

- Split graph
- Burkard–Hammer condition
- Burkard–Hammer graph
- Hamiltonian graph
- Maximal nonhamiltonian split graph