## The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs.(English)Zbl 1164.05356

Summary: If $$G$$  is a connected graph of order $$n \geq 1$$, then by a hamiltonian coloring of  $$G$$ we mean a mapping  $$c$$ of $$V(G)$$ into the set of all positive integers such that $$| c(x) - c(y)| \geq n - 1 - D_{G}(x, y)$$ (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 $$n \geq 3$$. Assume that there exists a subgraph  $$F$$ of  $$G$$ such that $$F$$  is a hamiltonian-connected graph of order  $$i$$, where $$2 \leq i \leq \frac 12(n + 1)$$. Then $$\text{hc}(G) \leq (n - 2)^2 + 1 - 2(i - 1)(i - 2)$$.

### MSC:

 05C15 Coloring of graphs and hypergraphs 05C38 Paths and cycles 05C45 Eulerian and Hamiltonian graphs 05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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### References:

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