Combinatorica最新文献

We prove that for all integers (Delta ,r ge 2), there is a constant (C = C(Delta ,r) >0) such that the following is true for every sequence ({mathcal {F}}= {F_1, F_2, ldots }) of graphs with (v(F_n) = n) and (Delta (F_n) le Delta ), for each (n in {mathbb {N}}). In every r-edge-coloured (K_n), there is a collection of at most C monochromatic copies from ({mathcal {F}}) whose vertex-sets partition (V(K_n)). This makes progress on a conjecture of Grinshpun and Sárközy.

An (alpha ,beta )-Kempe swap in a properly colored graph interchanges the colors on some component of the subgraph induced by colors (alpha ) and (beta ). Two k-colorings of a graph are k-Kempe equivalent if we can form one from the other by a sequence of Kempe swaps (never using more than k colors). Las Vergnas and Meyniel showed that if a graph is ((k-1))-degenerate, then each pair of its k-colorings are k-Kempe equivalent. Mohar conjectured the same conclusion for connected k-regular graphs. This was proved for (k=3) by Feghali, Johnson, and Paulusma (with a single exception (K_2square ,K_3), also called the 3-prism) and for (kge 4) by Bonamy, Bousquet, Feghali, and Johnson. In this paper we prove an analogous result for list-coloring. For a list-assignment L and an L-coloring (varphi ), a Kempe swap is called L-valid for (varphi ) if performing the Kempe swap yields another L-coloring. Two L-colorings are called L-equivalent if we can form one from the other by a sequence of L-valid Kempe swaps. Let G be a connected k-regular graph with (kge 3) and (Gne K_{k+1}). We prove that if L is a k-assignment, then all L-colorings are L-equivalent (again excluding only (K_2square ,K_3)). Further, if (Gin {K_{k+1},K_2square ,K_3}), L is a (Delta )-assignment, but L is not identical everywhere, then all L-colorings of G are L-equivalent. When (kge 4), the proof is completely self-contained, implying an alternate proof of the result of Bonamy et al. Our proofs rely on the following key lemma, which may be of independent interest. Let H be a graph such that for every degree-assignment (L_H) all (L_H)-colorings are (L_H)-equivalent. If G is a connected graph that contains H as an induced subgraph, then for every degree-assignment (L_G) for G all (L_G)-colorings are (L_G)-equivalent.

We give a short constructive proof for the existence of a Hamilton cycle in the subgraph of the ((2n+1))-dimensional hypercube induced by all vertices with exactly n or (n+1) many 1s.

Let A be a subset of the cyclic group ({textbf{Z}}/p{textbf{Z}}) with p prime. It is a well-studied problem to determine how small |A| can be if there is no unique sum in (A+A), meaning that for every two elements (a_1,a_2in A), there exist (a_1',a_2'in A) such that (a_1+a_2=a_1'+a_2') and ({a_1,a_2}ne {a_1',a_2'}). Let m(p) be the size of a smallest subset of ({textbf{Z}}/p{textbf{Z}}) with no unique sum. The previous best known bounds are (log p ll m(p)ll sqrt{p}). In this paper we improve both the upper and lower bounds to (omega (p)log p leqslant m(p)ll (log p)^2) for some function (omega (p)) which tends to infinity as (prightarrow infty ). In particular, this shows that for any (Bsubset {textbf{Z}}/p{textbf{Z}}) of size (|B|<omega (p)log p), its sumset (B+B) contains a unique sum. We also obtain corresponding bounds on the size of the smallest subset of a general Abelian group having no unique sum.

A sweep of a point configuration is any ordered partition induced by a linear functional. Posets of sweeps of planar point configurations were formalized and abstracted by Goodman and Pollack under the theory of allowable sequences of permutations. We introduce two generalizations that model posets of sweeps of higher dimensional configurations. Sweeps of a point configuration are in bijection with faces of an associated sweep polytope. Mimicking the fact that sweep polytopes are projections of permutahedra, we define sweep oriented matroids as strong maps of the braid oriented matroid. Allowable sequences are then the sweep oriented matroids of rank 2, and many of their properties extend to higher rank. We show strong ties between sweep oriented matroids and both modular hyperplanes and Dilworth truncations from (unoriented) matroid theory. Pseudo-sweeps are a generalization of sweeps in which the sweeping hyperplane is allowed to slightly change direction, and that can be extended to arbitrary oriented matroids in terms of cellular strings. We prove that for sweepable oriented matroids, sweep oriented matroids provide a sphere that is a deformation retract of the poset of pseudo-sweeps. This generalizes a property of sweep polytopes (which can be interpreted as monotone path polytopes of zonotopes), and solves a special case of the strong Generalized Baues Problem for cellular strings. A second generalization are allowable graphs of permutations: symmetric sets of permutations pairwise connected by allowable sequences. They have the structure of acycloids and include sweep oriented matroids.

We show that if a finite point set (Psubseteq {mathbb {R}}^2) has the fewest congruence classes of triangles possible, up to a constant M, then at least one of the following holds.

• There is a (sigma >0) and a line l which contains (Omega (|P|^sigma )) points of P. Further, a positive proportion of P is covered by lines parallel to l each containing (Omega (|P|^sigma )) points of P.

• There is a circle (gamma ) which contains a positive proportion of P.

This provides evidence for two conjectures of Erdős. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.

Let (n ge 2k ge 4) be integers, ({[n]atopwithdelims ()k}) the collection of k-subsets of ([n] = {1, ldots , n}). Two families ({mathcal {F}}, {mathcal {G}} subset {[n]atopwithdelims ()k}) are said to be cross-intersecting if (F cap G ne emptyset ) for all (F in {mathcal {F}}) and (G in {mathcal {G}}). A family is called non-trivial if the intersection of all its members is empty. The best possible bound (|{mathcal {F}}| + |{mathcal {G}}| le {n atopwithdelims ()k} - 2 {n - katopwithdelims ()k} + {n - 2k atopwithdelims ()k} + 2) is established under the assumption that ({mathcal {F}}) and ({mathcal {G}}) are non-trivial and cross-intersecting. For the proof a strengthened version of the so-called shifting technique is introduced. The most general result is Theorem 4.1.

The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by Babai (J Graph Theory 9:363–370, 1985). Then Babai posed the open problems of constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose adjacency matrices are not diagonalizable. In this paper, we solve Babai’s problems by constructing an infinite family of s-arc-transitive digraphs for each integer (sge 2), and an infinite family of vertex-primitive digraphs, both of whose adjacency matrices are non-diagonalizable.

In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent.

Let ({mathbb {F}}_q) be a finite field of characteristic p and order q. The Chevalley–Warning Theorem asserts that the set V of common zeros of a collection of polynomials must satisfy (|V|equiv 0mod p), provided the number of variables is sufficiently large with respect to the degrees of the polynomials. The Ax–Katz Theorem generalizes this by giving tight bounds for higher order p-divisibility for |V|. Besides the intrinsic algebraic interest of these results, they are also important tools in the Polynomial Method, particularly in the prime field case ({mathbb {F}}_p), where they have been used to prove many results in Combinatorial Number Theory. In this paper, we begin by explaining how arguments used by Wilson to give an elementary proof of the ({mathbb {F}}_p) case for the Ax–Katz Theorem can also be used to prove the following generalization of the Ax–Katz Theorem for ({mathbb {F}}_p), and thus also the Chevalley–Warning Theorem, where we allow varying prime power moduli. Given any box ({mathcal {B}}={mathcal {I}}_1times ldots times {mathcal {I}}_n), with each ({mathcal {I}}_jsubseteq {mathbb {Z}}) a complete system of residues modulo p, and a collection of nonzero polynomials (f_1,ldots ,f_sin {mathbb {Z}}[X_1,ldots ,X_n]), then the set of common zeros inside the box,

satisfies (|V|equiv 0mod p^m), provided (n>(m-1)max _{iin [1,s]}Big {p^{m_i-1}deg f_iBig }+ sum nolimits _{i=1}^{s}frac{p^{m_i}-1}{p-1}deg f_i.) The introduction of the box ({mathcal {B}}) adds a degree of flexibility, in comparison to prior work of Sun. Indeed, incorporating the ideas of Sun, a weighted version of the above result is given. We continue by explaining how the added flexibility, combined with an appropriate use of Hensel’s Lemma to choose the complete system of residues ({mathcal {I}}_j), allows many combinatorial applications of the Chevalley–Warning and Ax–Katz Theorems, previously only valid for ({mathbb {F}}_p^n), to extend with bare minimal modification to validity for an arbitrary finite abelian p-group G. We illustrate this by giving several examples, including a new proof of the exact value of the Davenport Constant ({textsf{D}}(G)) for finite abelian p-groups, and a streamlined proof of the Kemnitz Conjecture. We also derive some new results, for a finite abelian p-group G with exponent q, regarding the constant ({textsf{s}}_{kq}(G)), defined as the minimal integer (ell ) such that any sequence of (ell ) terms from G must contain a zero-sum subsequence of length kq.