On the smallest snarks with oddness 4 and connectivity 2

A snark is a bridgeless cubic graph which is not 3-edge-colourable. The oddness of a bridgeless cubic graph is the minimum number of odd components in any 2-factor of the graph. Lukot'ka, M\'acajov\'a, Maz\'ak and \v{S}koviera showed in [Electron. J. Combin. 22 (2015)] that the smallest snark with oddness 4 has 28 vertices and remarked that there are exactly two such graphs of that order. However, this remark is incorrect as -- using an exhaustive computer search -- we show that there are in fact three snarks with oddness 4 on 28 vertices. In this note we present the missing snark and also determine all snarks with oddness 4 up to 34 vertices.Comment: 5 page

Using "tangibles" to promote novel forms of playful learning

Tangibles, in the form of physical artefacts that are electronically augmented and enhanced to trigger various digital events to happen, have the potential for providing innovative ways for children to play and learn, through novel forms of interacting and discovering. They offer, too, the scope for bringing playfulness back into learning. To this end, we designed an adventure game, where pairs of children have to discover as much as they can about a virtual imaginary creature called the Snark, through collaboratively interacting with a suite of tangibles. Underlying the design of the tangibles is a variety of transforms, which the children have to understand and reflect upon in order to make the Snark come alive and show itself in a variety of morphological and synaesthesic forms. The paper also reports on the findings of a study of the Snark game and discusses what it means to be engrossed in playful learning

Snark Wars

The latest volley in the war of words waged by cultured despisers of Christianity was fired on Christmas Day. Celebrity astrophysicist Neil deGrasse Tyson, host of the television series Cosmos, bushwhacked Christians with this tweeted broadside: On this day long ago, a child was born who, by age 30, would transform the world. Happy Birthday Isaac Newton b. Dec 25, 1642. Not content with just one shot, Tyson let fly again. Merry Christmas to all, he tweeted. A Pagan holiday (BC) becomes a Religious holiday (AD). Which then becomes a Shopping holiday (USA). Then, the coup de grace. QUESTION: This year, what do all the world\u27s Muslims and Jews call December 25th? ANSWER: Thursday. [excerpt

Generation of cubic graphs and snarks with large girth

We describe two new algorithms for the generation of all non-isomorphic cubic graphs with girth at least $k\ge 5$ which are very efficient for $5\le k \le 7$ and show how these algorithms can be efficiently restricted to generate snarks with girth at least $k$. Our implementation of these algorithms is more than 30, respectively 40 times faster than the previously fastest generator for cubic graphs with girth at least 6 and 7, respectively. Using these generators we have also generated all non-isomorphic snarks with girth at least 6 up to 38 vertices and show that there are no snarks with girth at least 7 up to 42 vertices. We present and analyse the new list of snarks with girth 6.Comment: 27 pages (including appendix

Some snarks are worse than others

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 graph which is not 3--edge-colourable. In this paper we deal with the fact that the family of potential counterexamples to many interesting conjectures can be narrowed even further to the family ${\cal S}_{\geq 5}$ of bridgeless cubic graphs whose edge set cannot be covered with four perfect matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover Conjecture and the Fan-Raspaud Conjecture are examples of statements for which ${\cal S}_{\geq 5}$ is crucial. In this paper, we study parameters which have the potential to further refine ${\cal S}_{\geq 5}$ and thus enlarge the set of cubic graphs for which the mentioned conjectures can be verified. We show that ${\cal S}_{\geq 5}$ can be naturally decomposed into subsets with increasing complexity, thereby producing a natural scale for proving these conjectures. More precisely, we consider the following parameters and questions: given a bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii) how many copies of the same perfect matching need to be added, and (iii) how many 2--factors need to be added so that the resulting regular graph is Class I? We present new results for these parameters and we also establish some strong relations between these problems and some long-standing conjectures.Comment: 27 pages, 16 figure