Geodesic
Hamiltonicity of cubic 3-connected \(k\)-Halin graphs
Shabnam Malik1 ; Ahmad Mahmood Qureshi1 ; Tudor Zamfirescu2
1Forman Christian College, Lahore, Pakistan
2Faculty of Mathematics Dortmund University of Technology 44221 Dortmund Germany
The electronic journal of combinatorics, Tome 20 (2013) no. 1
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We investigate here how far we can extend the notion of a Halin graph such that hamiltonicity is preserved. Let $H = T \cup C$ be a Halin graph, $T$ being a tree and $C$ the outer cycle. A $k$-Halin graph $G$ can be obtained from $H$ by adding edges while keeping planarity, joining vertices of $H - C$, such that $G - C$ has at most $k$ cycles. We prove that, in the class of cubic $3$-connected graphs, all $14$-Halin graphs are hamiltonian and all $7$-Halin graphs are $1$-edge hamiltonian. These results are best possible.
DOI : 10.37236/3188
Classification : 05C45, 05C38
Mots-clés : k-Halin graph, Hamiltonian cycles, k-edge Hamiltonian

Shabnam Malik  1   ; Ahmad Mahmood Qureshi  1   ; Tudor Zamfirescu  2

1 Forman Christian College, Lahore, Pakistan
2 Faculty of Mathematics Dortmund University of Technology 44221 Dortmund Germany 
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Shabnam Malik; Ahmad Mahmood Qureshi; Tudor Zamfirescu. Hamiltonicity of cubic 3-connected \(k\)-Halin graphs. The electronic journal of combinatorics, Tome 20 (2013) no. 1. doi: 10.37236/3188

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