Combinatorics (math.CO)

Abstract: Algebraic Combinatorics originated in Algebra and Representation Theory, studying their discrete objects and integral quantities via combinatorial methods which have since developed independent and self-contained lives and brought us some beautiful formulas and combinatorial interpretations. The flagship hook-length formula counts the number of Standard Young Tableaux, which also gives the dimension of the irreducible Specht modules of the Symmetric group. The elegant Littlewood-Richardson rule gives the multiplicities of irreducible GL-modules in the tensor products of GL-modules. Such formulas and rules have inspired large areas of study and development beyond Algebra and Combinatorics, becoming applicable to Integrable Probability and Statistical Mechanics, and Computational Complexity Theory. We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity Theory, that could formally explain the beauty we see and the difficulties we encounter in finding further formulas and ``combinatorial interpretations''. A 85-year-old such problem asks for a positive combinatorial formula for the Kronecker coefficients of the Symmetric group, another one pertains to the plethysm coefficients of the General Linear group. In the opposite direction, the study of Kronecker and plethysm coefficients leads to the disproof of the wishful approach of Geometric Complexity Theory (GCT) towards the resolution of the algebraic P vs NP Millennium problem, the VP vs VNP problem. In order to make GCT work and establish computational complexity lower bounds, we need to understand representation theoretic multiplicities in further detail, possibly asymptotically.

Abstract: We show that the chronological removal of cycles from a walk on a graph, known as Lawler's loop-erasing procedure, generates a preLie co-algebra on the vector space spanned by the walks. In addition, we prove that the tensor and symmetric algebras of graph walks are graded Hopf algebras, provide their antipodes explicitly and recover the preLie co-algebra from a brace coalgebra on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras associated to particular types of walks.

Abstract: The $k$-edge-colored bipartite Gallai-Ramsey number $\operatorname{bgr}_k(G:H)$ is defined as the minimum integer $n$ such that $n^2\geq k$ and for every $N\geq n$, every edge-coloring (using all $k$ colors) of complete bipartite graph $K_{N,N}$ contains a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we first study the structural theorem on the complete bipartite graph $K_{n,n}$ with no rainbow copy of $K_{1,3}$. Next, we utilize the results to prove the exact values of $\operatorname{bgr}_{k}(P_4: H)$, $\operatorname{bgr}_{k}(P_5: H)$, $\operatorname{bgr}_{k}(K_{1,3}: H)$, where $H$ is a various union of cycles and paths and stars.

Abstract: We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $\Delta(G) + 1$ colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$ of subexponential growth can be properly edge-colored using $\Delta(G) + 1$ colors by an $O(\log^\ast n)$-round deterministic distributed algorithm in the $\mathsf{LOCAL}$ model, where the implied constants in the $O(\cdot)$ notation are determined by a bound on the growth rate of $G$.

Abstract: We use the link between Jacobi continued fractions and the generating functions of certain moment sequences to study some simple transformations on them. In particular, we define and study a transformation that is appropriate for the study of spidernet graphs and their moments, and the free Meixner law.

Abstract: How many copies of a fixed odd cycle, $C_{2m+1}$, can a planar graph contain? We answer this question asymptotically for $m\in\{2,3,4\}$ and prove a bound which is tight up to a factor of $3/2$ for all other values of $m$. This extends the prior results of Cox--Martin and Lv et al. on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass $\mu$ on the edges of some clique maximizes the probability that $m$ edges sampled independently from $\mu$ form either a cycle or a path?