Eulerian Glued Graphs

V. Boonthong, P. Putthapiban, P. Chaisuriya, P. Pacheenburawana

Abstract

The Glued Graph is the graph that results from combining two graphs by overlapping their subgraphs with respect to some isomorphism. In this paper we show that the glued graph of any two nontrivial connected graphs is Eulerian if and only if the following conditions hold:
1.) the clones of the two original graphs contain all odd vertices from their two original graphs,
2.) every even vertex in the clones of the two original graphs is obtained from both odd or both even vertices in the two original graphs, and every odd vertex in those clones is obtained from one odd and one even vertex in the two original graphs.
Moreover, the glued graph of two Eulerian graphs is also Eulerian if and only if the clones are Eulerian. In addition, we show that the glued graph of two connected graphs, where one of these is Eulerian and one is not, is Eulerian if and only if the following conditions hold:
1.) the clone of the non-Eulerian graph contains all odd vertices,
2.) every even vertex in the clones of the two original graphs is obtained from both even vertices in two original graphs.