On cycles in regular 3-partite tournaments

A $c$-partite tournament is an orientation of a complete $c$-partite graph. In 2006, Volkmann conjectured that every arc of a regular 3-partite tournament $D$ is contained in an $m$-, $(m+1)$- or $(m+2)$-cycle for each $m\in \{3,4,\dots ,\left|V\left(D\right)\right|?2\}$, and this conjecture was proved to be correct for $3\le m\le 7$. In 2012, Xu et al. conjectured that every arc of an $r$-regular 3-partite tournament $D$ with $r\ge 2$ is contained in a $(3k?1)$- or $3k$-cycle for $k=2,3,\dots ,r$. They proved that this conjecture is true for $k=2$. In this paper, we confirm this conjecture for $k=3$, which also implies that Volkmann’s conjecture is correct for $m=7,8$.     

Classifying (almost)-Belyi maps with five exceptional points

We classify all rational functions $f:{\mathbb{P}}^1\to {\mathbb{P}}^1$ whose branching pattern above $0,1,\infty$ satisfy a certain regularity condition with precisely $d=5$ exceptions. This work is motivated by solving second order linear differential equations, with $d=5$ true singularities, in terms of hypergeometric functions. A similar problem was solved for $d=4$ in Vidunas and Filipuk (2013).     

Distinguishing index of maps

The distinguishing number of a group $A$ acting on a finite set $\Omega$, denoted by $D(A,\Omega )$, is the least $k$ such that there is a $k$-coloring of $\Omega$ which is preserved only by elements of $A$ fixing all points in $\Omega$. For a map $M$, also called a cellular graph embedding or ribbon graph, the action of $Aut\left(M\right)$ on the vertex set $V$ gives the distinguishing number $D\left(M\right)$. It is known that $D\left(M\right)\le 2$ whenever $\left|V\right|>10$. The action of $Aut\left(M\right)$ on the edge set $E$ gives the distinguishing index $D^{\prime }\left(M\right)$, which has not been studied before. It is shown that the only maps $M$ with $D^{\prime }\left(M\right)>2$ are the following: the tetrahedron; the maps in the sphere with underlying graphs $C_n$, or $K_{1,n}$ for $n=3,4,5$; a map in the projective plane with underlying graph $C_4$; two one-vertex maps with 4 or 5 edges; one two-vertex map with 4 edges; or any map obtained from these maps using duality or Petrie duality. There are 39 maps in all.     

No-three-in-line problem on a torus: Periodicity

Let ${\tau }_{m,n}$ denote the maximal number of points on the discrete torus (discrete toric grid) of sizes $m\times n$ with no three collinear points. The value ${\tau }_{m,n}$ is known for the case where $gcd(m,n)$ is prime. It is also known that ${\tau }_{m,n}\le 2gcd(m,n)$. In this paper we generalize some of the known tools for determining ${\tau }_{m,n}$ and also show some new. Using these tools we prove that the sequence ${\left({\tau }_{z,n}\right)}_{n\in \mathbb{N}}$ is periodic for all fixed $z>1$. In general, we do not know the period; however, if $z=p^a$ for $p$ prime, then we can bound it. We prove that ${\tau }_{p^a,p^{(a?1)p+2}}=2p^a$ which implies that the period for the sequence is $p^b$, where $b$ is at most $(a?1)p+2$.     

Largest regular multigraphs with three distinct eigenvalues

We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are largest. For $k\ne 2,3,7,57$, we show an upper bound $n\le k^2?k+1$, with equality if and only if there exists a finite projective plane of order $k?1$ that admits a polarity.     

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.     

Subgroup regular sets in Cayley graphs

Let Γ be a graph with vertex set V, and let a and b be nonnegative integers. A subset C of V is called an $(a,b)$-regular set in Γ if every vertex in C has exactly a neighbors in C and every vertex in $V?C$ has exactly b neighbors in C. In particular, $(0,1)$-regular sets and $(1,1)$-regular sets in Γ are called perfect codes and total perfect codes in Γ, respectively. A subset C of a group G is said to be an $(a,b)$-regular set of G if there exists a Cayley graph of G which admits C as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group G or any group G of order 4p or pq for some primes p and q, if a nontrivial subgroup H of G is a $(0,1)$-regular set of G, then it must also be an $(a,b)$-regular set of G for any $0?a?|H|?1$ and $0?b?|H|$ such that a is even when $|H|$ is odd. A similar result involving $(1,1)$-regular sets of such groups is also obtained in the paper.     

Factoring periodic maps into Dehn twists

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\ge 1$. In this paper, we develop various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy. As applications, we develop methods for factoring certain roots of Dehn twists as words in Dehn twists. We will also show the existence of conjugates of periodic maps of order 4g and $4g+2$, for $g\ge 2$, whose product is pseudo-Anosov.     

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 $k\ge 2$.