Let G be a nontrivial connected graph of order n. For u, v ∈ V(G), the detour distance D(u, v) between u and v is the length of a longest u - v path in G. A hamiltonian coloring of G is an assignment c of colors (positive integers) to the vertices of G that satisfies D(u, v) + |c(u) - c(v)| ≥ n - 1 for every two distinct vertices u and v of G. The value of a hamiltonian coloring c of G is hc(c) = max{c(v) : v ∈ V(G)}. The hamiltonian chromatic number hc(G) of G is the minimum value hc(c) among all hamiltonian colorings c of G. We provide bounds for the hamiltonian chromatic number of a graph in terms of its order and detour distances ir the graph.
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机译：令G为n阶的平凡连通图。对于u，v∈V（G），u和v之间的de回距离D（u，v）是G中最长的u-v路径的长度。G的哈密顿着色是颜色的赋值c（正整数） ）到满足D（u，v）+ | c（u）-c（v）|的G的顶点对于G的每两个不同的顶点u和v≥n-1。G的哈密顿着色c的值为hc（c）= max {c（v）：v∈V（G）}。 G的哈密顿色数hc（G）是G的所有哈密顿色数c中的最小值hc（c）。根据图的哈密顿色数的阶数和de回距离，我们提供了该图的哈密顿色数的界限。
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