On hamiltonian colorings of graphs

Hamiltonian Colorings of Graphs with Long Cycles

By a hamiltonian coloring of a connected graph G of order n > 1 we mean a mapping c of V (G) into the set of all positive integers such that |c(x) − c(y)| > n − 1 − DG(x, y) (where DG(x, y) denotes the length of a longest x − y path in G) for all distinct x, y ∈ G. In this paper we study hamiltonian colorings of non-hamiltonian connected graphs with long cycles, mainly of connected graphs of or...

Perfect $2$-colorings of the Platonic graphs

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

On Irregular Colorings of Graphs

For a graph G and a proper coloring c : V (G) → {1, 2, . . . , k} of the vertices of G for some positive integer k , the color code of a vertex v of G (with respect to c ) is the ordered (k + 1) -tuple code(v) = (a0, a1, . . . , ak) where a0 is the color assigned to v and for 1 ≤ i ≤ k , ai is the number of vertices adjacent to v that are colored i . The coloring c is irregular if distinct vert...

On Total Colorings of Graphs

On multiset colorings of graphs

A vertex coloring of a graph G is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum k for which G has a multiset k-coloring is the multiset chromatic number χm(G) of G. For every graph G, χm(G) is bounded above by its chromatic number χ(G). The multiset chromatic numbers of regular graphs are investigated. It is shown that ...