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We conjecture that for any fixed r and sufficiently large n , there is a monochromatic Hamiltonian Berge-cycle in every ( r − 1 ) -coloring of the edges of K n ( r ) , the complete r -uniform hypergraph on n vertices. We prove the conjecture for r = 3 , n ⩾ 5 and its asymptotic version for r...
This paper considers instability of graphs in all of its possible forms. First, four theorems (one with two interesting special cases) are presented, each of which shows that graphs satisfying certain conditions are unstable. Several infinite families of graphs are investigated. For each family,...
We show that the complete bipartite graph K n , n has a unique regular embedding in an orientable surface if and only if n is coprime to φ ( n ) . The method, involving groups which factorise as a product of two cyclic groups, is also used to classify such embeddings when n is the square of a...
The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that k -colorability of a graph G is equivalent to the condition 1 ∈ I G , k for a certain ideal I G , k ⊆ k ( x 1 , … , x n ) . In...
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobás–Riordan–Tutte polynomial generalizes the Tutte polynomial of graphs to graphs that are embedded...
A graph G is pseudo 2-factor isomorphic if the parity of the number of circuits in a 2-factor is the same for all 2-factors of G . We prove that there exist no pseudo 2-factor isomorphic k -regular bipartite graphs for k ⩾ 4 . We also propose a characterization for 3-edge-connected pseudo...
A graph is prismatic if for every triangle T , every vertex not in T has exactly one neighbour in T . In a previous paper we gave a complete description of all 3-colourable prismatic graphs, and of a slightly more general class, the “orientable” prismatic graphs. In this paper we describe the...
We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this catalogue. Our computation confirms and extends the results obtained in the 1960s by Blackburn, Crapo and Higgs. The matroids and associated data are...
A d -dimensional box is a Cartesian product of d closed intervals on the real line. The boxicity of a graph is the minimum dimension d such that it is representable as the intersection graph of d -dimensional boxes. We give a short constructive proof that every graph with maximum degree D has...
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