EVEN FACTOR WITH SPECIFIED EDGES OF GRAPHS

The minimum order of components of a graph G is denoted by σ(G). Every 2-edge-connected graph of minimum degree at least 3 has an even factor containing two arbitrary prescribed edges. Jackson and Yoshimoto showed that if these graphs is simple, then there is an even factor F in which σ(F) ≥ 4. We prove that there is an even factor F containing two given edges such that σ(F) ≥ 4. They also showed that if G is a 3- edge-connected graph with | G | ≥ 5, v is a vertex of degree 3, e = vx and f = vy ∈ E(G), then G has an even factor F containing e and f in which σ(F) ≥ 5. We extend this result and prove that this theorem is satisfied for each pair of adjacent edges and every 3-edge-connected graph has an even factor F containing two given edges e and f such that every component containing neither e nor f has order at least 5. But, we construct infinitely many 3-edge-connected graphs that do not have an even factor F containing two arbitrary prescribed edges in which σ(F) ≥ 5.