594 research outputs found
Distinguishing graphs by their left and right homomorphism profiles
We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte
uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu.
We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are
specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and
Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from
multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and
Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky.
Unifying the study of these and related problems is the notion of the left and right homomorphism profiles
of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164
Monochromatic triangles in three-coloured graphs
In 1959, Goodman determined the minimum number of monochromatic triangles in
a complete graph whose edge set is two-coloured. Goodman also raised the
question of proving analogous results for complete graphs whose edge sets are
coloured with more than two colours. In this paper, we determine the minimum
number of monochromatic triangles and the colourings which achieve this minimum
in a sufficiently large three-coloured complete graph.Comment: Some data needed to verify the proof can be found at
http://www.math.cmu.edu/users/jcumming/ckpsty
The Zeta Function of a Hypergraph
We generalize the Ihara-Selberg zeta function to hypergraphs in a natural
way. Hashimoto's factorization results for biregular bipartite graphs apply,
leading to exact factorizations. For -regular hypergraphs, we show that
a modified Riemann hypothesis is true if and only if the hypergraph is
Ramanujan in the sense of Winnie Li and Patrick Sol\'e. Finally, we give an
example to show how the generalized zeta function can be applied to graphs to
distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.Comment: 24 pages, 6 figure
Complexity and Heegaard genus of an infinite class of compact 3-manifolds
Using the theory of hyperbolic manifolds with totally geodesic boundary, we
provide for every integer n greater than 1 a class of such manifolds all having
Matveev complexity equal to n and Heegaard genus equal to n+1. All the elements
of this class have a single boundary component of genus n, and the numbers of
distinct members of the class grows at least exponentially with n.Comment: 15 pages, 7 figure
Recent developments in graph Ramsey theory
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress
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