On snarks that are far from being 3-edge colorable

On measures of edge-uncolorability of cubic graphs: A brief survey and some new results

There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called {\em snarks}, play a key role in this context. Here, we survey parameters measuring how far apart a non 3-edge-colorable graph is from being 3-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we…

Cyclic pseudo-{L}oupekine snarks

In 1976, Loupekine introduced (via Isaacs) a very general way of constructing new snarks from old snarks by cyclically connecting multipoles constructed from smaller snarks. In this paper, we generalize Loupekine's construction to produce a variety of snarks which can be drawn with $m$-fold rotational symmetry for $m\geq 3$ (and often, $m$ odd), constructed as $\mathbb{Z}_{m}$ lifts of \emph{voltage graphs} with certain properties; we call these snarks \emph{cyclic pseudo-Loupekine snarks}. In particular, we discuss three infinite families of snarks which can be drawn with $\mathbb{Z}_{m}$ rot…

循环5-边连通的电阻为2且流电阻为n的snarks

Cyclically $5$-edge-connected snarks with resistance $2$ and flow resistance $n$

本文构造了一类循环5-边连通的snarks，其流电阻与电阻的比值可任意大，回答了Allie等人提出的问题。

Quest for graphs of Frank number $3$

In an orientation $O$ of the graph $G$, the edge $e$ is deletable if and only if $O-e$ is strongly connected. For a $3$-edge-connected graph $G$, Hörsch and Szigeti defined the Frank number as the minimum $k$ for which $G$ admits $k$ orientations such that every edge $e$ of $G$ is deletable in at least one of the $k$ orientations. They conjectured the Frank number is at most $3$ for every $3$-edge-connected graph $G$. They proved the Petersen graph has Frank number $3$, but this was the only example with this property. We show an infinite class of graphs having Frank number $3$. Hörsch and Szi…

Some results on the structure of multipoles in the study of snarks

Multipoles are the pieces we obtain by cutting some edges of a cubic graph. As a result of the cut, a multipole $M$ has dangling edges with one free end, which we call semiedges. Then, every 3-edge-coloring of a multipole induces a coloring or state of its semiedges, which satisfies the Parity Lemma. Multipoles have been extensively used in the study of snarks, that is, cubic graphs which are not 3-edge-colorable. Some results on the states and structure of the so-called color complete and color closed multipoles are presented. In particular, we give lower and upper linear bounds on the minimu…

围长或循环连通度等于6的严格临界snark图

Strictly critical snarks with girth or cyclic connectivity equal to 6

针对循环连通度大于5的严格临界snark图研究空白，本文构造了各偶数阶n≥342的循环6-连通严格临界snark图，还构造了围长6、满足n≡2(mod8)且n≥66的循环5-连通严格临界snark图。