13,791 research outputs found
A note on the history of the four-colour conjecture
The four-colour conjecture was brought to public attention in 1854, most
probably by Francis or Frederick Guthrie. This moves back by six years the date
of the earliest known publication.Comment: 3 pages; revised sourcing, added extra informatio
Recursive generation of IPR fullerenes
We describe a new construction algorithm for the recursive generation of all
non-isomorphic IPR fullerenes. Unlike previous algorithms, the new algorithm
stays entirely within the class of IPR fullerenes, that is: every IPR fullerene
is constructed by expanding a smaller IPR fullerene unless it belongs to
limited class of irreducible IPR fullerenes that can easily be made separately.
The class of irreducible IPR fullerenes consists of 36 fullerenes with up to
112 vertices and 4 infinite families of nanotube fullerenes. Our implementation
of this algorithm is faster than other generators for IPR fullerenes and we
used it to compute all IPR fullerenes up to 400 vertices.Comment: 19 pages; to appear in Journal of Mathematical Chemistr
Switching Reconstruction of Digraphs
Switching about a vertex in a digraph means to reverse the direction of every
edge incident with that vertex. Bondy and Mercier introduced the problem of
whether a digraph can be reconstructed up to isomorphism from the multiset of
isomorphism types of digraphs obtained by switching about each vertex. Since
the largest known non-reconstructible oriented graphs have 8 vertices, it is
natural to ask whether there are any larger non-reconstructible graphs. In this
paper we continue the investigation of this question. We find that there are
exactly 44 non-reconstructible oriented graphs whose underlying undirected
graphs have maximum degree at most 2. We also determine the full set of
switching-stable oriented graphs, which are those graphs for which all
switchings return a digraph isomorphic to the original
Fullerenes with distant pentagons
For each , we find all the smallest fullerenes for which the least
distance between two pentagons is . We also show that for each there is
an such that fullerenes with pentagons at least distance apart and
any number of hexagons greater than or equal to exist.
We also determine the number of fullerenes where the minimum distance between
any two pentagons is at least , for , up to 400 vertices.Comment: 15 pages, submitted for publication. arXiv admin note: text overlap
with arXiv:1501.0268
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