A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Many combinatorial problems can be efficiently solved for partial k-trees (graphs of treewidth bounded by a constant k). However, no polynomial-time algorithm has been known for the problem of finding a total coloring of a giv...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is sdegenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤ s. We prove that an s-degenerate graph G has a total coloring with ∆+1 colors if the maximum degre...

introduction: to assess the salivary composition of proteins and minerals in smokers compared with non-smokers.   materials and methods: in this study we compared the total protein and ca, na, k, mg, pb of whole saliva in two groups of men (28 smokers and 31nonsmokers) aged between 29-41years.   results: fifty-nine participants were evaluated. the mean age was 33.14±5.32 years among smokers and...

A set S of vertices in a graph G = (V,E) is called a total k-distance dominating set if every vertex in V is within distance k of a vertex in S. A graph G is total k-distance domination-critical if γ t (G − x) < γ t (G) for any vertex x ∈ V (G). In this paper, we investigate some results on total k-distance domination-critical of graphs.

Let k be a positive integer and G be a k-connected graph. In 2009, Chartrand, Johns, McKeon, and Zhang introduced the rainbow k-connection number rck(G) of G. An edge-coloured path is rainbow if its edges have distinct colours. Then, rck(G) is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths...