On Hamiltonian colorings of graphs. (English) Zbl 1059.05046

The authors give a lower bound for the circumference of a graph in terms of the number of vertices that receive colors between two specified colors in a Hamiltonian coloring of the graph. As a consequence, if there exists a Hamiltonian coloring of a connected graph \(G\) of order \(n\) such that at least \((n+2)/2\) vertices of \(G\) are colored with one of two consecutive colors, then the circumference of \(G\) is at least \(n-1\).


05C15 Coloring of graphs and hypergraphs
05C12 Distance in graphs
05C38 Paths and cycles
05C45 Eulerian and Hamiltonian graphs
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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