Interval Edge Coloring Of Several Special Graphs

Given a graph G,we need to give a proper edge coloring of the G,as for this coloring,each of the vertex of G that incident to edges has different colors,those colors are consecutive or those color could be an interval of integers,so we will say this graph G is an interval colorable graph.This topic comes from a situation:the school holds parenting meeting,they want to arrange the time in advance and the meeting time should be fixed for each group,where all the teachers attending the meeting constitute a set of A,all the parents constitute a set of B,so the set of A and B can be two vertices of a bipartite graph.If a teacher and a parent need to meet,the two vertices can add one edge,then using the integer numbers to color the graph,then the number of colors on each edge is the agreed meeting time.The school hopes to either the parent or the teacher will be able to go straight to the next meeting after a meeting,without wasting time to wait for the next meeting.After arranging,it is found that the colors of each point of the graph is continuous.Given a graph,it is a very difficult task to determine whether this graph has an interval edge coloring.In fact,someone has been proved that if given a bipartite graph that is interval colorable graph,which is a NP-complete problem.In this topic,we mainly describe the relation information about it,and for?a,b?-biregular bipartite graph,in the second chapter,we get some results:we give a sufficient condition of?3,4?-biregular bipartite graph has an interval edge coloring and some results about?a,ka?-,?a,3a?-,?4,4k?-biregular bipartite graph.Further more,there are some people have interest in the tripartite graph,in the third chapter,we mainly have some results about several special kinds of complete tripartite graph such as K_2,m,m,K_l,m,k?l+m?etc.In the last chapter,we describe the future work and the problems to be solved.