On hamiltonian colorings of graphs

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

“…Let G 1 be a connected factor of a graph G 0 . As immediately follows from Lemma 4.5 in [2], hc(G 0 ) hc(G 1 ). This result is easy but very useful.…”

“…The notions of a hamiltonian coloring and the hamiltonian chromatic number of a connected graph were introduced by Chartrand, Nebeský and Zhang in [2]. The adjective "hamiltonian" in these terms has a transparent motivation: if G is a connected graph, then hc(G) = 1 if and only if G is hamiltonian-connected.…”

“…The connected graph of order n which is, in a very natural sense, the most different from the hamiltonian-connected graphs of order n is the star K 1,n−1 . It was proved in [2] that hc(K 1,n−1 ) = (n − 2) 2 + 1. As was proved in [3], if G is a connected graph of order n 5 which is not a star, then hc(G) hc(K 1,n−1 ) − 2.…”

“…It was proved in [2] that hc(K 1,n−1 ) = (n − 2) 2 + 1. As was proved in [3], if G is a connected graph of order n 5 which is not a star, then hc(G) hc(K 1,n−1 ) − 2. As follows from another result proved in [2], hc(C n ) = hc(K 1,n−1 ) − 1 = n − 2.…”

On the Detour and Vertex Detour Hull Numbers of a Graph

“…The detour number of a graph was introduced in [3] and further studied in [4,8]. These concepts have interesting applications in Channel Assignment Problem in radio technologies [5,7]. A set S of vertices of G is a detour hull set if [S] D = V (G) and a detour hull set of minimum cardinality is the detour hull number dh(G).…”