Weak Vertex-edge Coloring Of Plane Graphs

For a plane graph G=(V,E,F),we use V,E,F to donate its vertex set,edge set and face set.In 2016,Fabrici,Jendrol' and Vrbjarova proposed the concept of weak vertex-edge coloring of plane graphs.If e1 and e1 are consecutively adjacent to the same face,then we say that e1 and e1 are facially adjacent.A plane graph G is called weakly vertex-edge k-colorable if there exists a mapping ?:V(G)?E(G)?{1,...,k},so that any two incident elements,adjacent vertices and facially adjacent edges receive distinct colors.The weak vertex-edge chromatic number of G,denoted by ?ve(G)is defined to be the least integer k such that G has a weak vertex-edge k-coloring.Fabrici,Jendrol' and Vrbjarova conjecture that every connected,loopless and bridgeless plane graph is weakly vertex-edge 5-colorable.This problem has attracted great interests of researchers.Up to now,this conjecture has not been completely solved.This thesis focuses on this type coloring problem.In other words,it is necessary for us to find some special plane graphs satisfying the conjecture.The structure and content of the thesis are as follows:In Chapter 1,we first give some basic concepts of graph theory which will be used throughout of thesis.Then,we briefly describe the research status of related fields,and finally present the main results of this thesis.In Chapter 2,we use mathematical induction and coloring extension methods to prove that every Halin graph is weakly vertex-edge 5-colorable.Moreover,we also find a special Halin graph,such as K4,whose weak vertex-edge chromatic number is exactly 5.So the upper bound 5 is the best possible.In Chapter 3,by characterizing the structural properties of 2-outerplanar graphs with ??5,and using the coloring permutation techniques,we prove that every 2-outerplanar graph with ??5 is weakly vertex-edge 5-colorable.