The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

Nebeský, Ladislav

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 317-338

Résumé

If $G$ is a connected graph of order $n \ge 1$, then by a hamiltonian coloring of $G$ we mean a mapping $c$ of $V(G)$ into the set of all positive integers such that $\vert c(x) - c(y)\vert \ge n - 1 - D_{G}(x, y)$ (where $D_{G}(x, y)$ denotes the length of a longest $x-y$ path in $G$) for all distinct $x, y \in V(G)$. Let $G$ be a connected graph. By the hamiltonian chromatic number of $G$ we mean \[ \min (\max (c(z);\, z \in V(G))), \] where the minimum is taken over all hamiltonian colorings $c$ of $G$. The main result of this paper can be formulated as follows: Let $G$ be a connected graph of order $n \ge 3$. Assume that there exists a subgraph $F$ of $G$ such that $F$ is a hamiltonian-connected graph of order $i$, where $2 \le i \le \frac{1}{2}(n + 1)$. Then $\mathop {\mathrm hc}(G) \le (n - 2)^2 + 1 - 2(i - 1)(i - 2)$.

Détail

Classification : 05C15, 05C38, 05C45, 05C78
Keywords: connected graphs; hamiltonian-connected subgraphs; hamiltonian colorings; hamiltonian chromatic number