Smarandachely Adjacent Vertex Distinguishing Total Coloring Of Outer Planar Graph

The coloring problem of graphs is one of the important problems in graph theory,which origi-nates from the well-known "four-color conjecture" problem.The coloring of graphs not only plays an important role in discrete mathematics,chemistry,computer and other fields,but also is widely used in real life.A[k]-total coloring of graph G is a mapping f:V(G)U E(G)[k]={1,2,…,k}such that any pair of adjacent or associated elements in V(G)U E(G)are colored differently.Let f[u]represents the color of vertex u and the set of colors of all edges associated with vertex u.A[k]-total coloring of graph G is called Smarandachely adjacent vertex distinguished,if for every edge of G,there exists |f[u]\f[v]| ≥ 1 and |f[u]\f[u]|≥ 1.In the Smarandachely adjacent vertex distinguishing total coloring of graph G,the smallest k is called Smarandachely adjacent vertex distinguishing total chromatic number of graph G,is denoted by χsat(G).In this paper,according to the structural characteristics of outer planar graphs,the Smaran-dachely adjacent vertex distinguishing total chromatic number problem of outerplanar graphs is main-ly studied by analysis and induction methods as well as techniques of color shifting,color changing and abstraction,divided into three chapters.The first chapter introduces the basic concepts,research background and research results on adjacent vertex distinguishable total coloring and Smarandachely adjacent vertex distinguishable total coloring.Finally,several conclusions about Smarandachely ad-jacent vertex distinguishable total coloring of outerplanar graphs are given.In Chapter 2,we study Smarandachely adjacent vertex-distinguishing panchromatic numbers of 2-connected outerplanar graphs with maxima of 3,4,5,respectively.It is proved that:If G is a 2-connectd outer planar graph with △(G)≤3,then χsat(G)≤△(G)+3;If G is a 2-connectd outer planar graph with △(G)=4 or 5,then χsat(G)≤△(G)+4.In Chapter 3,we study Smarandachely adjacent vertex distinguish-able panchromatic number of outerplanar graphs and give the upper bound of χsat(G)not exceeding 2△(G);furthermore,if G is an outer planar graph of △(G)≥ 7,then χsat(G)≤2△(G)-1.