2020/08/09 by Edita Máčajová, Máčajová, Edita, Martin Škoviera +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2008.04775
openalex publication_date 2020/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The perfect matching index of a cubic graph G, denoted by π(G), is the smallest number of perfect matchings that cover all the edges of G. According to the Berge-Fulkerson conjecture, π(G)≤5 for every bridgeless cubic graph~G. The class of graphs with π≥ 5 is of particular interest as many conjectures and open problems, including the famous cycle double cover conjecture, can be reduced to it. Although nontrivial examples of such graphs are very difficult to find, a few infinite families are known, all with circular flow number Φc(G)=5. It has been therefore suggested [Electron. J. Combin. 23 (2016), #P3.54] that π(G)≥ 5 might imply Φc(G)≥ 5. In this article we dispel these hopes and present a family of cyclically 4-edge-connected cubic graphs of girth at least 5 (snarks) with π≥ 5 and Φc≤ 4+\frac23.