A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge.A plane graph with near independent crossings(say NIC-planar graph) is a 1-planar graph with the restriction that for any two crossings the four crossed edges are incident with at most one common vertex.The full characterization of NIC-planar complete and complete multipartite graphs is given in this paper.
An edge coloring totalk-labeling is a labeling of the vertices and the edges of a graph G with labels{1,2,...,k}such that the weights of the edges defne a proper edge coloring of G.Here the weight of an edge is the sum of its label and the labels of its two end vertices.This concept was introduce by Brandt et al.They defnedχt(G)to be the smallest integer k for which G has an edge coloring total k-labeling and proposed a question:Is there a constant K withχt(G)≤Δ(G)+12+K for all graphs G of maximum degreeΔ(G)?In this paper,we give a positive answer for outerplanar graphs by showing thatχt(G)≤Δ(G)+12+1 for each outerplanar graph G with maximum degreeΔ(G).
The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ'' (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ'' (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ''(G) = 9.
In this paper,we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group(Δ(G)+1)-edge-choosable,and some planar graphs with large girth and maximum degree are groupΔ(G)-edge-choosable.
A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h]={1,2,...,h}.Letw(u) denote the sum of the color on a vertex u and colors on all the edges incident to u.For each edge uv∈E(G),if w(u)≠w(v),then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G.By tndi(G),we denote the smallest value h in such a coloring of G.In this paper,we obtain that G is a graph with at least two vertices,if mad(G)<3,then tndi∑(G)≤k+2 where k=max{Δ(G),5}.It partially con?rms the conjecture proposed by Pil′sniak and Wozniak.
