Supereulerian graphs: A survey

Abstract

A graph is supereulerian if it has a spanning eulerian subgraph. There is a reduction method to determine whether a graph is supereulerian, and it can also be applied to study other concepts, e.g., hamiltonian line graphs, a certain type of double cycle cover, and the total interval number of a graph. We outline the research on supereulerian graphs, the reduction method and its applications.

On the minimal length of the longest trail in a fixed edge-density graph

“…The method of the proof is to find a large Eulerian circuit in a subgraph of G and to construct from it a long enough trail in G. A similar problem, finding spanning circuits, has been studied extensively in [1]. In order to keep the article self-contained, we give all propositions with proofs.…”

Connected Factors in Graphs ? a Survey

“…[31] surveyed supereulerian graphs, i.e., graphs containing a connected even factor. Zelinka [153] showed that for every integer r there exists a 2-edge connected graph G, of order n, with no Eulerian subgraph covering more than n/r vertices of G. One important result about supereulerian graphs is the following.…”

“…Zhang [154], the survey of Lesniak and Oellermann [95] or that of Catlin [31]. 2) If a = 2 and n ≥ 3, δ(G) ≥ max{3,…”

Supereulerian graphs and the Petersen graph

“…Pulleyblank [16] showed that determining if a graph is supereulerian, even when restricted to planar graphs, is NP-complete. For more in the literature on supereulerian graphs, see Catlin's survey [6] and its update by Chen and Lai [11].…”