Using the existence of noncrossing Eulerian circuits in Eulerian plane graphs, we give a short constructive proof of the theorem of Heawood that Eulerian triangulations are 3-colorable.

A graph is supereulerian if it has a spanning Eulerian subgraph. Motivated by the Chinese Postman Problem, Boesch, Suffel, and Tindell ([2]) in 1997 proposed the supereulerian problem, which seeks a characterization of graphs that have spanning Eulerian subgraphs, and they indicated that this problem would be very difficult. Pulleyblank ([71]) later in 1979 proved that determining whether a gra...

A graph is subeulerian if it is spanned by an eulerian supergraph. Boesch, Suffel and Tindell have characterized the class of subeulerian graphs and determined the minimum number of additional lines required to make a subeulerian graph eulerian. In this paper, we consider the related notion of a subsemi-eulerian graph, i.e. a graph which is spanned by a supergraph having an open trail containin...

We characterize the graphs that admit a decomposition into circuits, i.e. finite or infinite connected 2-regular graphs. Moreover, we show that, as is the case for the removal of a closed eulerian subgraph from a finite graph, removal of a non-dominated eulerian subgraph from a (finite or infinite) graph does not change its circuit-decomposability or circuit-indecomposability. For cycle-decompo...

We give counterexamples to two conjectures in \Bill Jackson, Some remarks on arc-connectivity, vertex splitting, and orientation in graphs and digraphs, Journal of Graph Theory, 12(3):429{436, 1988" concerning orientations of mixed graphs and splitting oo in digraphs, and prove the rst conjecture in the (di-) Eulerian case(s). Beside that we solve the degree constrained directed augmentation pr...