Edge Connectivity And Ultra-eulerian Graph

An eulerian graph is a connected graph with no odd vertices .A graph is supereulerianif it has a spanning eulerian subgraph. Eulerian graph problem is one of the most classicproblems in the graph theory.While the study of supereulerian graph is an important hotproblem of Eulerian graph theory,it is also a NP-complete problem. This dissertation mainlyinvestigated the relationship between edge-connectivity and supereulerian graph,especiallyfor the supereulerian studies of 2-edge-connected graph C(l,k).Using P.A.Catlin's contrac-tion method, we gained some important results about C(7,4),C(8,3) and C(9,2) on thebasis of earlier results .The dissertation consists of three chapters.In the first chapter,we introduced an overviewon the history of the development of graph theory, the background issues, as well as researchsituations and also pointed out the work of this dissertation.In the second chapter, the basic concepts,terminologies and P.A.Catlin's sub-graphcontraction method are intorducded and also pointed out the property of supereulerianwhen the edge-connectivity of graph is greater than or equal 3.In the last chapter,we specifically proved that if G belongs to C(l, k) (7≤l≤9,2≤l≤4) with F(G) less than three and under other conditions, then G is a supereulerian graphif and only if G cannot be contracted to any special graphs.