Abstract
The classical Berge–Fulkerson conjecture states that any bridgeless cubic graph G admits a list of six perfect matchings such that each edge of G belongs to two of the perfect matchings from the list. In this short note, we discuss two statements that are consequences of this conjecture. The first of them states that for any bridgeless cubic graph G, edge e and i with 0≤i≤2, there are three perfect matchings F1,F2,F3 of G such that F1∩F2∩F3=0̸ and e belongs to exactly i of these perfect matchings. The second one states that for any bridgeless cubic graph G and its vertex v, there are three perfect matchings F1,F2,F3 of G such that F1∩F2∩F3=0̸ and the edges incident to v belong to k1, k2 and k3 of these perfect matchings, where the numbers k1, k2 and k3 satisfy the obvious necessary conditions. In the paper, we show that the first statement is equivalent to Fan–Raspaud conjecture. We also show that the smallest counter-example to the second one is a cyclically 4-edge-connected cubic graph.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: AKCE International Journal of Graphs and Combinatorics
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
