Related papers: Group divisible designs with block size 4 and type…

Let $G=(V(G), E(G))$ be an undirected graph with a measure function $\mu$ assigning non-negative values to subgraphs $H$ so that $\mu(H)$ does not exceed the clique cover number of $H$. When $\mu$ satisfies some additional natural…

Requiring that supersymmetric SU(5) and SU(6) grand unifications in the heterotic string theory must have 3 chiral families, adjoint (or higher representation) Higgs fields in the grand unifiedgauge group, and a non-abelian hidden sector,…

We study the question of whether it is possible to determine a finitely generated group $G$ up to some notion of equivalence from the spectrum $\mathrm{sp}(G)$ of $G$. We show that the answer is "No" in a strong sense. As the first example…

We introduce and study the problem of finding necessary and sufficient conditions under which a conformal blocks divisor on $\bar{M}_{0,n}$ is nonzero. We give necessary conditions in type A, which are sufficient when theta and critical…

The Glashow-Salam-Weinberg model for N=2 generations is extended to 8 composite Higgs multiplets by using a one-to-one correspondence between its complex Higgs doublet and very specific quadruplets of bilinear quark operators. This is the…

For homogeneous reductive spaces G/H with reductive complements decomposable into an orthogonal sum \mathfrak{m}=\mathfrak{m}_1 \oplus \mathfrak{m}_2 \oplus \mathfrak{m}_3 of three Ad(H)-invariant irreducible mutually inequivalent…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex…

In orbifold gauge theories localized tadpoles can be radiatively generated at the fixed points where U(1) subgroups are conserved. If the Standard Model Higgs fields are identified with internal components of the bulk gauge fields…

The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star…

A graph $G$ is well-covered if all its maximal stable sets have the same size, denoted by alpha(G) (M. D. Plummer, 1970). If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$…

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. It is easy…

Let $S=\{x_1,\ldots, x_n\}$ be a gcd-closed set (i.e. $(x_i,x_j)\in S $ for all $1\le i,j\le n$). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets $S$ with $|S|\ge 4$ such that the GCD matrix $(S)$…

A cyclic subgroup $N$ of a finite group $G$ is called a uni-width subgroup of $G$ if $N$ is the unique cyclic subgroup of $G$ of order $|N|$. In this article, we prove that a finite group $G$ admits a unique largest uni-width subgroup…

In the framework of the four dimensional heterotic superstring with free fermions we discuss the rank eight and/or sixteen Grand Unified String Theories (GUST) which contain the SU(3)_H - gauge family symmetry. We explicitly investigate the…

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms…

There are six orientable, compact, flat 3-manifolds that can occur as cusp cross-sections of hyperbolic 4-manifolds. This paper provides criteria for exactly when a given commensurability class of arithmetic hyperbolic 4-manifolds contains…

This paper investigates $2$-$(v,5,\lambda)$ designs $\mathcal{D}$ admitting a block-transitive automorphism group $G$. We first prove that if $G$ is point-imprimitive, then $v$ must be one of 16, 21, or 81. We further provide a complete…

We prove that the Herzog-Sch\"onheim Conjecture holds for any group $G$ of order smaller than $1440$. In other words we show that in any non-trivial coset partition $\{g_i U_i\}_{i=1}^n $ of $G$ there exist distinct $1 \leq i, j \leq n$…

In this paper, we resolve a conjecture of Green and Liebeck [Disc. Math., 343 (8):117119, 2019] on codes in $PGL(2,q)$. To be specific, we show that: if $D$ is a dihedral subgroup of order $2(q+1)$ in $G=PGL(2,q)$, and $A=\{g\in G: g^{q+1}=…

Let $\Sigma_{g,b}$ denote a closed orientable surface of genus $g$ with $b$ punctures and let $\rm Mod(\Sigma_{\textit{g,b}})$ denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, $\rm…