Catlin'S Reduced Graphs With Small Orders

A graph is supereulerian if it has a spanning closed trail. Catlin in 1990 raised the problem of determining the reduced nonsupereulerian graphs with small orders, as such results are of particular importance in the study of Eulerian subgraphs and Hamiltonian line graphs. We determine all reduced graphs with order at most 14 and with few vertices of degree 2, extending former results of Chen and Chen in 2016. In 1985, Bauer proposed the problems of determining best possible sufficient conditions on minimum degree of a simple graph (or a simple bipartite graph, respectively) G to ensure that its line graph L(G) is Hamiltonian. These problems have been settled by Catlin and Lai in 1988, respectively. As an application of our main results, we prove the following for a connected simple graph G on n vertices:i. If doGTHORN delta(G)>= n/10, then for sufficiently large n, L(G) is Hamilton-connected if and only if both joGTHORN kappa (G) >= 3 and G is not nontrivially contractible to the Wagner graph. ii. If G is bipartite and doGTHORN delta(G) > n /20, then for sufficiently large n, L(G) is Hamilton-connected if and only if both joGTHORN K(G) >= 3 and G is not nontrivially contractible to the Wagner graph.