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It was proved by (M.R. Garey, D.S. Johnson, Crossing number is NP-complete, SIAM J. Algebraic Discrete Methods 4 (1983) 312–316) that computing the crossing number of a graph is an NP-hard problem. Their reduction, however, used parallel edges and vertices of very high degrees. We prove here...
Hajós conjectured that, for any positive integer k , every graph containing no K k + 1 -subdivision is k -colorable. This is true when k ⩽ 3 , and false when k ⩾ 6 . Hajós' conjecture remains open for k = 4 , 5 . In this paper, we show that any possible counterexample to this conjecture for...
A ( δ , γ ) -net in a matroid M is a pair ( N , P ) where N is a minor of M , P is a set of series classes in N , | P | ⩾ δ , and the pairwise connectivity, in M , between any two members of P is at least γ . We prove that, for any finite field F , nets provide a qualitative...
We construct a polynomial-time algorithm to approximate the branch-width of certain symmetric submodular functions, and give two applications. The first is to graph “clique-width.” Clique-width is a measure of the difficulty of decomposing a graph in a kind of tree-structure, and if a graph...
The cocircuit graph of an oriented matroid is the 1-skeleton of the cellular decomposition induced by the Topological Representation Theorem due to Folkman and Lawrence (1978) (J. Folkman, J. Lawrence, Oriented matroids, J. Combin. Theory Ser. B 25 (1978) 199–236). In this paper we exhibit a...
Thomassen conjectured that every 4-connected line graph is Hamiltonian. A vertex cut X of G is essential if G − X has at least two non-trivial components. We prove that every 3-connected, essentially 11-connected line graph is Hamiltonian. Using Ryjáček's line graph closure, it follows that...
The regularity lemma for 3-uniform hypergraphs asserts that every large hypergraph can be decomposed into a bounded number of quasi-random structures consisting of a sub-hypergraph and a sparse underlying graph. In this paper we show that in such a quasi-random structure most pairs of the edges...
Kuipers and Veldman conjectured that any 3-connected claw-free graph with order ν and minimum degree δ ⩾ ( ν + 6 ) 10 is Hamiltonian for ν sufficiently large. In this paper, we prove that if H is a 3-connected claw-free graph with sufficiently large order ν , and if δ ( H ) ⩾ ( ν + 5 )...
Let G be a graph with n vertices and m edges and assume that f : V ( G ) → N is a function with ∑ v ∈ V ( G ) f ( v ) = m + n . We show that, if we can assign to any vertex v of G a list L v of size f ( v ) such that G has a unique vertex coloring with these lists, then G is f -choosable....
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