In this note we present House of Graphs (http://hog.grinvin.org) which is a new database of graphs. The key principle is to have a searchable database and offer -next to complete lists of some graph classes -also a list of special graphs that already turned out to be interesting and relevant in the study of graph theoretic problems or as counterexamples to conjectures. This list can be extended by users of the database.
For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for snarks, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured.In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on n ≤ 36 vertices. Previously lists up to n = 28 vertices have been published.In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.
We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes - fullgen - and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for 136 or more vertices. We tabulate the numbers of fullerenes and IPR fullerenes up to 400 vertices. We also check up to 316 vertices a conjecture of Barnette that cubic planar graphs with maximum face size 6 are Hamiltonian and verify that the smallest counterexample to the spiral conjecture has 380 vertices.
Using computational techniques we derive six new upper bounds on the classical twocolor Ramsey numbers: R(3, 10) ≤ 42, R(3, 11) ≤ 50, R(3, 13) ≤ 68, R(3, 14) ≤ 77, R(3, 15) ≤ 87, and R(3, 16) ≤ 98. All of them are improvements by one over the previously best published bounds.Let e(3, k, n) denote the minimum number of edges in any triangle-free graph on n vertices without independent sets of order k. The new upper bounds on R(3, k) are obtained by completing the computation of the exact values of e(3, k, n) for all n with k ≤ 9 and for all n ≤ 33 for k = 10, and by establishing new lower bounds on e(3, k, n) for most of the open cases for 10 ≤ k ≤ 15. The enumeration of all graphs witnessing the values of e(3, k, n) is completed for all cases with k ≤ 9. We prove that the known critical graph for R(3, 9) on 35 vertices is unique up to isomorphism. For the case of R(3, 10), first we establish that R(3, 10) = 43 if and only if e(3, 10, 42) = 189, or equivalently, that if R(3, 10) = 43 then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that R(3, 10) ≤ 42.
Discrete Algorithms International audience We describe a new algorithm for the efficient generation of all non-isomorphic connected cubic graphs. Our implementation of this algorithm is more than 4 times faster than previous generators. The generation can also be efficiently restricted to cubic graphs with girth at least 4 or 5.
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