Collapsible subgraphs of a 4-edge-connected graph.

Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, graphs that have spanning eulerian subgraphs. Catlin in 1988 sharpened Jaeger’s result by showing that every 4-edge-connected graph is collapsible, graphs that are contractible configurations of supereulerian graphs. To further study collapsible subgraphs of a 4-edge-connected graph, in Catlin et al. (2009), it is shown that every 4-edge-connected graph remains collapsible after removing any two edges. We prove the following. •[(i)] Every 4-edge-connected G contains two vertices x,y such that one of x and y has minimum degree in G and both G−x and G−y are collapsible.•[(ii)] Let G be a 4-edge-connected graph and let X⊂E(G) be an edge subset with |X|≤3. Then G−X is collapsible if and only if X is not contained in a 4-edge-cut of G.•[(iii)] Let G be a 4-edge-connected graph and let X⊂E(G) be an edge subset with |X|≤4. Then G−X is collapsible if and only if G−X is not contractible to a member in {K2c,K2,K2,2,K2,3,K2,4}. These extend former results of Jaeger (1979) and Catlin (1988).