Paths and cycles concerning independent edges

Abstract

R. Häggkvist and C. Thomassen defined admissible paths in [5] which is a generalization of alternating paths. A path or a cycleD is said to beadmissible for a setL of pairwise independent edges if each edge ofL lies onD or has no vertex in common withD. For given two distinct verticesa andb, and a given setL of independent edges inG, we establish a necessary and sufficient condition for the existence of an admissible path forL connectinga andb inG. This theorem is analogous to Tutte's Theorem of Alternating Connection [14]. Futhermore we show a 2-connected version of this theorem which does not seem to have any analogue in the “theory of alternating path”. As a corollary of this theorem, we prove the 1-factor theorem. We conjecture that for a setL of given independent edges, |L| ≧n, in ann-connected graphG, with very few exceptionsG has an admissible cycleC forL such thatC contains at leastn edges ofL. Using our results concerning admissible paths, we also prove this conjecture in the cases wheren = 2 and 3. Furthermore, we show how our results are applied to problems of alternating paths. As a final application, we prove that ifG is a 3-regular graph with a Hamiltonian cycleH, then for any edgee ∈ E(H), there exists an admissible cycle forE(G) — E(H) inG through the edgee.