A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2 V （K1 ∪ K2） with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1 V （K1∪K2） with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.
A graph is 1-toroidal, if it can be embedded in the torus so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-toroidal graph with maximum degree △ ≥ 10 is of class one in terms of edge coloring. Meanwhile, we show that there exist class two 1-toroidal graphs with maximum degree △ for each A ≤ 8.
An edge coloring total k-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 define 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 defined Xt＇（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 with X^t（G） ≤△（G）＋1/2 K for all graphs G of maximum degree A（G）？ In this paper, we give a positive answer for outerplanar graphs ≤△（G）＋1/2 by showing that X＇t（G） ≤△（G）＋1/2 for each outerplanar graph G with maximum degree A（G）.
Let G=(V,E)be a graph andφbe a total coloring of G by using the color set{1,2,...,k}.Let f(v)denote the sum of the color of the vertex v and the colors of all incident edges of v.We say thatφis neighbor sum distinguishing if for each edge uv∈E(G),f(u)=f(v).The smallest number k is called the neighbor sum distinguishing total chromatic number,denoted byχ′′nsd(G).Pil′sniak and Wo′zniak conjectured that for any graph G with at least two vertices,χ′′nsd(G)(G)+3.In this paper,by using the famous Combinatorial Nullstellensatz,we show thatχ′′nsd(G)2(G)+col(G)-1,where col(G)is the coloring number of G.Moreover,we prove this assertion in its list version.
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}. Let w（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{△（G）, 5}. It partially confirms the conjecture proposed by Pilgniak and Wolniak.
