A Turán problem on digraphs avoiding distinct walks of a given length with the same endpoints

Let $k\ge 4$ and $n\ge k+4$ be positive integers. We determine the maximum size of digraphs of order $n$ that avoid distinct walks of length $k$ with the same endpoints. We also characterize the extremal digraphs attaining this maximum number when $k\ge 5$.     

Rainbow spanning tree decompositions in complete graphs colored by cyclic 1-factorizations

Brualdi and Hollingsworth conjectured in Brualdi and Hollingsworth (1996) that in any complete graph $K_{2n}$, $n\ge 3$, which is properly colored with $2n?1$ colors, the edge set can be partitioned into $n$ edge disjoint rainbow spanning trees (where a graph is said to be rainbow if its edges have distinct colors). Constantine (2002) strengthened this conjecture asking the rainbow spanning trees to be pairwise isomorphic. He also showed an example satisfying his conjecture for every $2n\in \{2^s:s\ge 3\}\cup \{5?2^s,s\ge 1\}$ . Caughmann, Krussel and Mahoney (2017) recently showed a first infinite family of edge colorings for which the conjecture of Brualdi and Hollingsworth can be verified. In the present paper, we extend this result to all edge-colorings arising from cyclic 1-factorizations of $K_{2n}$ constructed by Hartman and Rosa (1985). Finally, we remark that our constructions permit to extend Constatine’s result also to all $2n\in \{2^{s}d:s\ge 1,d>3\text{odd}\}$.     

A new phase transition in the parabolic Anderson model with partially duplicated potential

We investigate a variant of the parabolic Anderson model, introduced in previous work, in which an i.i.d. potential is partially duplicated in a symmetric way about the origin, with each potential value duplicated independently with a certain probability. In previous work we established a phase transition for this model on the integers in the case of Pareto distributed potential with parameter $\alpha >1$ and fixed duplication probability $p\in (0,1)$: if $\alpha \ge 2$ the model completely localises, whereas if $\alpha \in (1,2)$ the model may localise on two sites. In this paper we prove a new phase transition in the case that $\alpha \ge 2$ is fixed but the duplication probability $p\left(n\right)$ varies with the distance from the origin. We identify a critical scale $p\left(n\right)\to 1$, depending on $\alpha$, below which the model completely localises and above which the model localises on exactly two sites. We further establish the behaviour of the model in the critical regime.     

Sobolev homeomorphisms are dense in volume preserving automorphisms

In this paper we prove a weak version of Lusin's theorem for the space of Sobolev-$(1,p)$ volume preserving homeomorphisms on closed and connected n-dimensional manifolds, $n\ge 3$, for $p<n?1$. We also prove that if $p>n$ this result is not true. More precisely, we obtain the density of Sobolev-$(1,p)$ homeomorphisms in the space of volume preserving automorphisms, for the weak topology. Furthermore, the regularization of an automorphism in a uniform ball centered at the identity can be done in a Sobolev-$(1,p)$ ball with the same radius centered at the identity.     

Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n\ge 2$, and on the simple associative superalgebras $M(m,n)$, $m,n\ge 1$, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from G-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be 0.     

Spanning trees with at most 4 leaves in K1,5-free graphs

In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with ${\sigma }_4\left(G\right)\ge n?1$ contains a spanning tree with at most 3 leaves. In this paper, we prove an analogue of Kyaw’s result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with ${\sigma }_5\left(G\right)\ge n?1$ contains a spanning tree with at most 4 leaves. Moreover, the degree sum condition “${\sigma }_5\left(G\right)\ge n?1$” is best possible.     

Vertex-disjoint rainbow triangles in edge-colored graphs

Let $G$ be an edge-colored graph of order $n$. The minimum color degree of $G$, denoted by ${\delta }^c\left(G\right)$, is the largest integer $k$ such that for every vertex $v$, there are at least $k$ distinct colors on edges incident to $v$. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if ${\delta }^c\left(G\right)\ge (n+1)∕2$, then $G$ contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if $n\ge 20$ and ${\delta }^c\left(G\right)\ge (n+2)∕2$, then $G$ contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if ${\delta }^c\left(G\right)\ge (n+k)∕2$, then $G$ contains $k$ vertex-disjoint rainbow triangles. For any integer $k\ge 2$, we show that if $n\ge 16k-12$ and ${\delta }^c\left(G\right)\ge n∕2+k-1$, then $G$ contains $k$ vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of $k$ edge-disjoint rainbow triangles.     

The higher-order differential operator for the generalized Jacobi polynomials — new representation and symmetry

For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by Koornwinder’s generalized Jacobi polynomials with four parameters $\alpha ,\beta \in {\mathbb{N}}_0$ and $M,N\ge 0$ determining the orthogonality measure on the interval $?1\le x\le 1$. The corresponding differential equation of order $2\alpha +2\beta +6$ is presented here as a linear combination of four elementary components which make the corresponding differential operator widely accessible for applications. In particular, we show that this operator is symmetric with respect to the underlying scalar product and thus verify the orthogonality of the eigenfunctions.     

Minimum supports of functions on the Hamming graphs with spectral constraints

We study functions defined on the vertices of the Hamming graphs $H(n,q)$. The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot i$ with corresponding eigenspaces $U_i(n,q)$ for $0\le i\le n$. In this work, we consider the problem of finding the minimum possible support (the number of nonzeros) of functions belonging to a direct sum $U_i(n,q)\oplus U_{i+1}(n,q)\oplus \cdots \oplus U_j(n,q)$ for $0\le i\le j\le n$. For the case $i+j\le n$ and $q\ge 3$ we find the minimum cardinality of the support of such functions and obtain a characterization of functions with the minimum cardinality of the support. In the case $i+j>n$ and $q\ge 4$ we also find the minimum cardinality of the support of functions, and obtain a characterization of functions with the minimum cardinality of the support for $i=j$, $i>\frac{n}{2}$ and $q\ge 5$. In particular, we characterize eigenfunctions from the eigenspace $U_i(n,q)$ with the minimum cardinality of the support for cases $i\le \frac{n}{2}$, $q\ge 3$ and $i>\frac{n}{2}$, $q\ge 5$.     

The space of lines in cyclic covers of projective space

Take positive integers m, n and d. Let Y be an m-fold cyclic cover of ${\mathbb{P}}^n$ ramified over a general hypersurface $X\subset P^n$ of degree md. In this paper we study the space $F(Y)$ of lines in Y and show that it is smooth of dimension $2(n-1)-d(m-1)$ if $md>2n-3$ and $2(n-1)-d(m-1)\ge 0$. When $2(n-1)=d(m-1)$, our result gives a formula on the number of m-contact order lines of X (see Definition 1.2).     

3-Vertices with fewest 2-neighbors in plane graphs with no long paths of 2-vertices

Let $g(k,t)$ be the minimum integer such that every plane graph with girth g at least $g(k,t)$, minimum degree $\delta =2$ and no $(k+1)$-paths consisting of vertices of degree 2, where $k\ge 1$, has a 3-vertex with at least t neighbors of degree 2, where $1\le t\le 3$.In 2015, Jendrol' and Maceková proved $g(1,1)\le 7$. Later on, Hudák et al. established $g(1,3)=10$, Jendrol', Maceková, Montassier, and Soták proved $g(1,1)\ge 7$, $g(1,2)=8$ and $g(2,2)\ge 11$, and we recently proved that $g(2,2)=11$ and $g(2,3)=14$.Thus $g(k,t)$ is already known for $k=1$ and all t. In this paper, we prove that $g(k,1)=3k+4$, $g(k,2)=3k+5$, and $g(k,3)=3k+8$ whenever $k\ge 2$.