On R-hued Coloring Of 2-connected Uterplanar Graph And Acyclic Edge Coloring Of 1-planar Graph Without Adjacent Triangles

A graph G is a pair G=(V, E), which contains two finite nonempty set V and E. Each element in set V is called vertex and each element in set E of 2-element subsets of V is called edge. The set V and E are the vertex set and edge set of G. A graph G is called simple graph which is not ring and multi-edges. No special instructions, simple, finite and undirected graphs are here.A planar graph G is called outerplanar if it has a plane embedding such that all vertices lie on the boundary of some face.A planar graph G is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge.A (k, r)-coloring of a graph G is a mapping c:V(G)→k satisfying both the following.(a) c(u)≠c(v) for every edge uv∈E(G);(b)|c(NG(v))|≥min{dG(v,r} for any v∈V(G).A proper k-edge coloring of G is called acyclic if there is no bichromatic cycles in G. The acyclic edge chromatic number xa(G) of G is the smallest integer k such that G is acyclically edge k-colorable.In this article, we mainly study on r-hued coloring of 2-connected outerplanar graph and acyclic edge coloring of 1-planar graph without adjacent triangles.In the first chapter, we show some basic notations in this paper and give the back-ground of interrelated filed and present.In the second chapter, we mainly proved the result about r-hued coloring of 2-connected outerplanar graph with order p> 2:(i) Xr(G)≤f(r)(ii) XL,r(G)≤fr)+1we define:In the third chapter, we mainly proved that acyclic edge coloring of 1-planar graph without adjacent triangles and xα’(G)≤Δ+30.In the fourth chapter, do summarize and give the prospects.