Article

Full entry | Fulltext not available (moving wall 24 months)
Keywords:
$\{K_{1,4},K_{1,4}+e\}$-free graph; neighborhood union; traceable
Summary:
A graph $G$ is called $\{H_1,H_2, \dots ,H_k\}$-free if $G$ contains no induced subgraph isomorphic to any graph $H_i$, $1\leq i\leq k$. We define $$\sigma _k= \min \biggl \{ \sum _{i=1}^k d(v_i) \colon \{v_1, \dots ,v_k\}\ \text {is an independent set of vertices in}\ G \biggr \}.$$ In this paper, we prove that (1) if $G$ is a connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and $\sigma _3(G)\geq n-1$, then $G$ is traceable, (2) if $G$ is a 2-connected $\{K_{1,4},K_{1,4}+e\}$-free graph of order $n$ and $|N(x_1)\cup N(x_2)|+|N(y_1)\cup N(y_2)|\geq n-1$ for any two distinct pairs of non-adjacent vertices $\{x_1,x_2\}$, $\{y_1,y_2\}$ of $G$, then $G$ is traceable, i.e., $G$ has a Hamilton path, where $K_{1,4}+e$ is a graph obtained by joining a pair of non-adjacent vertices in a $K_{1,4}$.
References:
[1] Bauer, D., Fan, G., Veldman, H. J.: Hamiltonian properties of graphs with large neighborhood unions. Discrete Math. 96 (1991), 33-49. DOI 10.1016/0012-365X(91)90468-H | MR 1139438 | Zbl 0741.05039
[2] Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications. American Elsevier Publishing, New York (1976). DOI 10.1007/978-1-349-03521-2 | MR 0411988 | Zbl 1226.05083
[3] Duan, F., Wang, G. P.: Note on the longest paths in {$\{K_{1,4},K_{1,4}+e\}$}-free graphs. Acta Math. Sin., Engl. Ser. 28 (2012), 2501-2506. DOI 10.1007/s10114-012-0459-7 | MR 2995196 | Zbl 1259.05091
[4] Li, R.: Hamiltonicity of 2-connected $\{K_{1,4},K_{1,4}+e\}$-free graphs. Discrete Math. 287 (2004), 69-76. DOI 10.1016/j.disc.2004.05.014 | MR 2094057 | Zbl 1052.05505
[5] Li, R., Schelp, R. H.: Hamiltonicity of {$\{K_{1,4},K_{1,4}+e\}$}-free graphs. Discrete Math. 245 (2002), 195-202. DOI 10.1016/S0012-365X(01)00141-8 | MR 1887938 | Zbl 0995.05086
[6] Lin, H., Wang, J.: Hamilton paths in {$\{K_{1,4},K_{1,4}+e\}$}-free graphs. Discrete Math. 308 (2008), 4280-4285. DOI 10.1016/j.disc.2007.08.051 | MR 2427760 | Zbl 1235.05075

Partner of