The notion of ends equivalence classes on the set of rays (one-way infinite paths) of a graph is one of the most studied topics in infinite graph theory. An introduction to this theory and basic results can be found in Halin [3]. Halin defined two rays to be equivalent if no finite set of vertices can separate an infinite part of the first ray from an infinite part of the second one. In particu...

A proper total weighting of a graph G is mapping φ which assigns to each vertex and edge real number as its weight so that for any uv , Σ e ∈ E ( v ) φ( )+φ( ≠ u ). k,k ')-list assignment L set k permissible weights ' weights. An -total with z V ∪ called ')-choosable if every there exists weighting. As strenghtening the well-known 1-2-3 conjecture, it was conjectured in [Wong Zhu, Total choosab...

A strong edge-coloring of a graph G is a function that assigns to each edge a color such that two edges within distance two apart must receive different colors. The minimum number of colors used in a strong edge-coloring is the strong chromatic index of G. Lih and Liu [14] proved that the strong chromatic index of a cubic Halin graph, other than two special graphs, is 6 or 7. It remains an open...

Let G be a connected graph. Let f be a proper k -coloring of G and Π = (R_1, R_2, . . . , R_k) bean ordered partition of V (G) into color classes. For any vertex v of G, define the color code c_Π(v) of v with respect to Π to be a k -tuple (d(v, R_1), d(v, R_2), . . . , d(v, R_k)), where d(v, R_i) is the min{d(v, x)|x ∈ R_i}. If distinct vertices have distinct color codes, then we call f a locat...