Related papers: Cyclic pseudo-{L}oupekine snarks

In a proper edge-coloring of a cubic graph an edge $uv$ is called poor or rich, if the set of colors of the edges incident to $u$ and $v$ contains exactly three or five colors, respectively. An edge-coloring of a graph is normal, if any…

The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this paper we prove that deciding whether this…

It is well-known that the circular flow number of a bridgeless cubic graph can be computed in terms of certain partitions of its vertex-set with prescribed properties. In the present paper, we first study some of these properties that turn…

Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for cubic graphs. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless cubic…

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. H$\ddot{a}$gglund constructed two graphs Blowup$(K_4, C)$ and Blowup$(Prism,…

Let $C_{k_1}, \ldots, C_{k_n}$ be cycles with $k_i\geq 2$ vertices ($1\le i\le n$). By attaching these $n$ cycles together in a linear order, we obtain a graph called a polygon chain. By attaching these $n$ cycles together in a cyclic…

For some time the Petersen graph has been the only known Snark with circular flow number $5$ (or more, as long as the assertion of Tutte's $5$-flow Conjecture is in doubt). Although infinitely many such snarks were presented eight years ago…

In this paper we show that many snarks have shortest cycle covers of length $\frac{4}{3}m+c$ for a constant $c$, where $m$ is the number of edges in the graph, in agreement with the conjecture that all snarks have shortest cycle covers of…

The purpose of this article is twofold. First we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain…

For each nonnegative integer m we show that any closed, oriented topological four-manifold with fundamental group Z_{4m+2} and odd intersection form, with possibly seven exceptions, either admits no smooth structure or admits infinitely…

In this paper we further our understanding of the structure of class two cubic graphs, or snarks, as they are commonly known. We do this by investigating their 3-critical subgraphs, or as we will call them, minimal conflicting subgraphs. We…

The circumference $c(G)$ of a graph $G$ is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically $4$-, $5$- and $6$-edge-connected cubic graphs with…

We construct minimal surfaces by stacking doubly periodic Scherk surfaces one above another and gluing them along their ends. It is previously known that the Karcher--Meeks--Rosenberg (KMR) doubly periodic minimal surfaces and Meeks' family…

We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched covers with $m$ a prime power, this extension provides new knot concordance…

This is a survey or exposition of a particular collection of results and open problems involving snarks --- simple "cubic" (3-valent) graphs for which, for nontrivial reasons, the edges cannot be 3-colored. The results and problems here are…

The triplication method for constructing strong starters in $Z_{3m}$ from starters in $Z_{m}$ (say, a starter of order 21 from a starter of order 7) was proposed by the authors in 2025. The method reduced construction of the particular…

A graph $G$ is pseudo 2--factor isomorphic if the parity of the number of cycles in a 2--factor is the same for all 2--factors of $G$. In \cite{ADJLS} we proved that pseudo 2--factor isomorphic $k$--regular bipartite graphs exist only for…

In a (proper) edge-coloring of a bridgeless cubic graph G an edge e is rich (resp. poor) if the number of colors of all edges incident to end-vertices of e is 5 (resp. 3). An edge-coloring of G is is normal if every edge of G is either rich…

In this article we develop a new way of systematically constructing infinitely many families of smooth subvarieties $X$ of any given dimension $m$, $m \geq 3$, and any given codimension in $\mathbb P^N$, embedded by complete subcanonical…

A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid…