Related papers: Algorithms for the Generation of Snarks
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…
A Hist in a cubic graph $G$ is a spanning tree $T$ which has only vertices of degree three and one. A snark with a Hist is called a Hist-snark, see \cite{HO}. We present several computer generated Hist-snarks which form generalizations of…
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…
A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a 2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an…
The aim of this paper is to classify all snarks up to order $36$ and explain the reasons of their uncolourability. The crucial part of our approach is a computer-assisted structural analysis of cyclically $5$-connected critical snarks,…
A normal 5-edge-coloring of a cubic graph is a coloring such that for every edge the number of distinct colors incident to its end-vertices is 3 or 5 (and not 4). The well known Petersen Coloring Conjecture is equivalent to the statement…
A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…
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 well-known 5-flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5-flow Conjecture can be reduced to cubic…
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…
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…
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…
A scramble on a connected multigraph is a collection of connected subgraphs that generalizes the notion of a bramble. The maximum order of a scramble, called the scramble number of a graph, was recently developed as a tool for lower…
Let $G$ be a bridgeless cubic graph. The \textit{resistance} of $G$, denoted $r(G)$, is the minimum number of edges which can be removed from $G$ in order to render 3-edge-colourability. The \textit{oddness} of $G$, denoted $\omega(G)$, is…
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,…
An edge e is normal in a proper edge-coloring of a cubic graph G if the number of distinct colors on four edges incident to e is 2 or 4: A normal edge-coloring of G is a proper edge-coloring in which every edge of G is normal. The Petersen…
The main aim of this paper is to solve the design spectrum problem for Tietze's graph, the two 18-vertex Blanusa snarks, the six snarks on 20 vertices (including the flower snark J5), the twenty snarks on 22 vertices (including the two…
Strongly chordal graphs are a subclass of chordal graphs. The interest in this subclass stems from the fact that many problems which are NP-complete for chordal graphs are solvable in polynomial time for this subclass. However, we are not…
A graph $G$ with four or more vertices is called bicritical if the removal of any pair of distinct vertices of $G$ results in a graph with a perfect matching. A bicritical graph is minimal if the deletion of each edge results in a…
A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree D has a strong edge coloring with at most 4D + 4 colors. We…