Combinatorics (math.CO)

Abstract: In a (proper) edge-coloring of a bridgeless cubic graph G an edge e is rich (resp. poor) if the number of colors of all edges incident to end-vertices of e is 5 (resp. 3). An edge-coloring of G is is normal if every edge of G is either rich or poor. In this paper we consider snarks ~G obtained by a simple superposition of edges and vertices of a cycle C in a snark G: For an even cycle C we show that a normal coloring of G can be extended to a normal coloring of ~G without changing colors of edges outside C in G: An interesting remark is that this is in general impossible for odd cycles, since the normal coloring of a Petersen graph P10 cannot be extended to a superposition of P10 on a 5-cycle without changing colors outside the 5-cycle. On the other hand, as our colorings of the superpositioned snarks introduce 18 or more poor edges, we are inclined to believe that every bridgeless cubic graph istinct from P10 has a normal coloring with at least one poor edge and possibly with at least 6 if we also exclude the Petersen graph with one vertex truncated.

Abstract: By using Sulanke-Xin continued fractions method, Xin proposed a recursion system to solve the Somos 4 Hankel determinant conjecture. We find Xin's recursion system indeed give a sufficient condition for $(\alpha, \beta)$ Somos $4$ sequences. This allows us to prove 4 conjectures of Barry on $(\alpha, \beta)$ Somos $4$ sequences in a unified way.

Abstract: A corner in a map is an edge-vertex-edge triple consisting of two distinct edges incident to the same vertex. A corneration is a set of corners that covers every arc of the map exactly once. Cornerations in a dart-transitive map generalize the notion of a cycle structure in a symmetric graph. In this paper, we study the cornerations (and associated structures) that are preserved by a vertex-transitive group of automorphisms of the map.

Abstract: The vertices of the permutohedron can be covered by one hyperplane in the $n$ dimensional affine space. We prove here that any set of hyperplanes that covers all the vertices of the permutohedron but one contains at least ${n\choose 2}$ hyperplanes. Our proof is based on a new variant of the Combinatorial Nullstellensatz.

Abstract: Recently, Dragani\'c, Munh\'a Correia, Sudakov and Yuster showed that every tournament on $(2+o(1))k^2$ vertices contains a $1$-subdivision of a transitive tournament on $k$ vertices, which is tight up to a constant factor. We prove a counterpart of their result for immersions. Let $f(k)$ be the smallest integer such that any tournament on at least $f(k)$ vertices must contain a $1$-immersion of a transitive tournament on $k$ vertices. We show that $f(k)=O(k)$, which is clearly tight up to a multiplicative factor. If one insists in finding an immersion of a complete directed graph on $k$ vertices then an extra condition on the tournament is necessary. Indeed, we show that every tournament with minimum out-degree at least $Ck$ must contain a $2$-immersion of a complete digraph on $k$ vertices. This is again tight up to the value of $C$ and tight on the order of the paths in the immersion.

Abstract: In 1933, Karol Borsuk asked whether each bounded set in the $n$-dimensional Euclidean space can be divided into $n$+1 parts of smaller diameter. Because it would not make sense otherwise, one usually assumes that he just forgot to require that the whole set contains at least two points. The hypothesis that the answer to that question is positive became famous under the name \emph{Borsuk's conjecture}. Counterexamples are known for any $n\ge 64$, since 2013. Let $\Lambda$ the (original, unscaled) Leech lattice, a now very well-known infinite discrete vector set in the 24-dimensional Euclidean space. The smallest norm of nonzero vectors in $\Lambda$ is $\sqrt{32}$. Let $M$ the set of the 196560 vectors in $\Lambda$ having this norm. For each $x \in M$, $-x$ is in $M$. Let $H$ the set of all subsets of $M$ that for each $x$ in $M$ contain either $x$ or $-x$. Each element of $H$ has the same diameter $d = \sqrt{96}$. For dimensions $n<24$ one can analogously construct respective $M_n$ and $H_n$ from laminated $n$-dimensional sublattices of $\Lambda$. For uniformity, let $\Lambda_{24}=\Lambda$, $M_{24} = M$ and $H_{24} = H$. If $M_n$ is divisible into at most $n+1$ parts of diameter below $d$ then this applies to all elements of $H_n$, too. On the other hand, the more the minimum number of parts of diameter below $d$ that $M_n$ can be divided into exceeds $n+1$, the more likely $H_n$ contains a counterexample to Borsuk's conjecture. For dimensions $n$ from 22 to 24, I could not divide $M_n$ into less than 28, 32 and 39 parts of diameter below $d$, resp. (I did not find any applicable result by others).

Abstract: In this paper we develop non-existence results for $m$-ovoids in the classical polar spaces $Q^-(2r+1,q), W(2r-1,q)$ and $H(2r,q^2)$ for $r>2$. In [4] a lower bound on $m$ for the existence of $m$-ovoids of $H(4,q^2)$ is found by using the connection between $m$-ovoids, two-character sets, and strongly regular graphs. This approach is generalized in [3] for the polar spaces $Q^-(2r+1,q), W(2r-1,q)$ and $H(2r,q^2)$, $r>2$. In [1] an improvement for the particular case $H(4,q^2)$ is obtained by exploiting the algebraic structure of the collinearity graph, and using the characterization of an $m$-ovoid as an intruiging set. In this paper, we use an approach based on geometrical and combinatorial arguments, inspired by the results from [10], to improve the bounds from [3].

Abstract: In this note we elaborate on some notions of surface area for discrete graphs which are closely related to the inverse degree. These notions then naturally lead to an associated connectivity measure of graphs and to the definition of a special class of large graphs, called social graphs, that might prove interesting for applications, among others, in computer science.

Abstract: We provide the first-ever calculation of the number of inequivalent monotone Boolean functions of 9 variables, which is equal to 789,204,635,842,035,040,527,740,846,300,252,680.