Combinatorics (math.CO)

Abstract: Jaeger, Linial, Payan, and Tarsi introduced the notion of $A$-connectivity for graphs in 1992, and proved a decomposition for cubic graphs from which $A$-connectivity follows for all 3-edge-connected graphs when $|A|\geq 6$. The concept of $A$-connectivity was generalized to signed graphs by Li, Luo, Ma, and Zhang in 2018 and they proved that all 4-edge-connected flow-admissible signed graphs are $A$-connected when $|A|\geq 4$ and $|A|\neq 5$. We prove that all 3-edge-connected flow-admissible signed graphs are $A$-connected when $|A|\geq 6$ and $|A|\neq 7$. Our proof is based on a decomposition that is a signed-graph analogue of the decomposition found by Jaeger et. al, and which may be of independent interest.

Abstract: Schubert polynomials form a basis of the polynomial ring. This basis and its structure constants have received extensive study. Recently, Pan and Yu initiated the study of top Lascoux polynomials. These polynomials form a basis of a subalgebra of the polynomial ring where each graded piece has finite dimension. This paper connects Schubert polynomials and top Lascoux polynomials via a simple operator. We use this connection to show these two bases share the same structure constants. We also translate several results on Schubert polynomials to top Lascoux polynomials, including combinatorial formulas for their monomial expansions and supports.

Abstract: Extending a result of Christiansen, we prove that every mutli-graph $G=(V,E)$ admits a proper edge colouring $\phi:E\to \{1,2,\dots\}$ which is local, that is, $\phi(e)\le \max\{d(x)+\pi(x),d(y)+\pi(y)\}$ for every edge $e$ with end-points $x,y\in V$, where $d(z)$ (resp.\ $\pi(z)$) denotes the degree of a vertex $z$ (resp.\ the maximum edge multiplicity at $z$). This is derived from a local version of the Fan Equation.

Abstract: Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we obtain the strong metric dimension of the prime ideal sum graph for various classes of Artinian non-local commutative rings.

Abstract: Let $d$ and $n$ be positive integers such that $d|n$. Let $[n]=\{1,2,\ldots,n\}$ and $T$ be an endofunction on $[n]$. A subset $W$ of $[n]$ of cardinality $n/d$ is said to be $d$-splitting if $W \cup TW \cup \cdots \cup T^{d-1}W =[n]$. Let $\sigma(d;T)$ denote the number of $d$-splitting subsets. If $\sigma(2;T)>0$, then we show that $\sigma(2;T)=g_T(-1)$, where $g_T(t)$ is the generating function for the number of $T$-invariant subsets of $[n]$. It is interesting to note that substituting a root of unity into a polynomial with integer coefficients has an enumerative meaning. More generally, let $g_T(t_1,\ldots,t_d)$ be the generating function for the number of $d$-flags of $T$-invariant subsets. We prove for certain endofunctions $T$, if $\sigma(d;T)>0$, then $\sigma(d;T)=g_T(\zeta,\zeta^2,\ldots,\zeta^d)$, where $\zeta$ is a primitive $d^{th}$ root of unity.

Abstract: The integer Carath\'eodory rank of a pointed rational cone $C$ is the smallest number $k$ such that every integer vector contained in $C$ is an integral non-negative combination of at most $k$ Hilbert basis elements. We investigate the integer Carath\'eodory rank of simplicial cones with respect to their multiplicity, i.e., the determinant of the integral generators of the cone. One of the main results states that simplicial cones with multiplicity bounded by five have the integral Carath\'eodory property, that is, the integer Carath\'eodory rank equals the dimension. Furthermore, we present a novel upper bound on the integer Carath\'eodory rank which depends on the dimension and the multiplicity. This bound improves upon the best known upper bound on the integer Carath\'eodory rank if the dimension exceeds the multiplicity. At last, we present special cones which have the integral Carath\'eodory property such as certain dual cones of Gorenstein cones.

Abstract: The families $\mathcal F_1\subseteq \binom{[n]}{k_1},\mathcal F_2\subseteq \binom{[n]}{k_2},\dots,\mathcal F_r\subseteq \binom{[n]}{k_r}$ are said to be cross-intersecting if $|F_i\cap F_j|\geq 1$ for any $1\leq i<j\leq r$ and $F_i\in \mathcal F_i$, $F_j\in\mathcal F_j$. Cross-intersecting families $\mathcal F_1,\mathcal F_2,\dots,\mathcal F_r$ are said to be non-empty if $\mathcal F_i\neq\emptyset$ for any $1\leq i\leq r$. This paper shows that if $\mathcal F_1\subseteq\binom{[n]}{k_1},\mathcal F_2\subseteq\binom{[n]}{k_2},\dots,\mathcal F_r\subseteq\binom{[n]}{k_r}$ are non-empty cross-intersecting families with $k_1\geq k_2\geq\cdots\geq k_r$ and $n\geq k_1+k_2$, then $\sum_{i=1}^{r}|\mathcal F_i|\leq\max\{\binom{n}{k_1}-\binom{n-k_r}{k_1}+\sum_{i=2}^{r}\binom{n-k_r}{k_i-k_r},\ \sum_{i=1}^{r}\binom{n-1}{k_i-1}\}$. This solves a problem posed by Shi, Frankl and Qian recently. The extremal families attaining the upper bounds are also characterized.

Abstract: As a natural generalization of permutations, transversals of Young diagrams play an important role in the study of pattern avoiding permutations. Let $\mathcal{T}_{\lambda}(\tau)$ and $\mathcal{ST}_{\lambda}(\tau)$ denote the set of $\tau$-avoiding transversals and $\tau$-avoiding symmetric transversals of a Young diagram $\lambda$, respectively. In this paper, we are mainly concerned with the distribution of the peak set and the valley set on standard Young tableaux and pattern avoiding transversals. In particular, by introducing Knuth transformations on standard Young tableaux, we prove that the peak set and the valley set are equidistributed on the standard Young tableaux of shape $\lambda/\mu$ for any skew diagram $\lambda/\mu$. The equidistribution enables us to show that the peak set is equidistributed over $\mathcal{T}_{\lambda}(12\cdots k\tau)$ (resp. $\mathcal{ST}_{\lambda}(12\cdots k \tau)$) and $\mathcal{T}_{\lambda}(k\cdots 21\tau) $ (resp. $\mathcal{ST}_{\lambda}(k\cdots 21\tau)$) for any Young diagram $\lambda$ and any permutation $\tau$ of $\{k+1, k+2, \ldots, k+m\}$ with $k,m\geq 1$. Our results are refinements of the result of Backelin-West-Xin which states that $|\mathcal{T}_{\lambda}(12\cdots k\tau)|=|\mathcal{T}_{\lambda}(k\cdots 21 \tau)|$ and the result of Bousquet-M\'elou and Steingr\'imsson which states that $|\mathcal{ST}_{\lambda}(12\cdots k \tau)|=|\mathcal{ST}_{\lambda}(k\cdots 21 \tau)|$.

Abstract: For every graph that is mimimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edge lengths are transcendental and algebraically independent. Here we compute the Galois group of all minimally rigid graphs that can be constructed from a single edge by repeated Henneberg 1-steps. It turns out that any such group is totally imprimitive, i.e., it is determined by all the partitions it preserves.

Abstract: Let $\Gamma$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Denote the degree of $\Gamma$ by $d$ and the edge Cheeger constant of $\Gamma$ by $\mathfrak{h}_\Gamma$. Assume that $\Gamma$ is undirected, non-bipartite and has finitely many vertices. We prove that the edge bipartiteness constant of $\Gamma$ is $\Omega({\mathfrak{h}_\Gamma}/{d})$, and the smallest eigenvalue of the normalized adjacency operator of $\Gamma$ is $-1 + \Omega({\mathfrak{h}_\Gamma^2}/{d^2})$. These answer in the affirmative a question of Moorman, Ralli and Tetali on the edge bipartiteness constant of Cayley graphs, and a question of them on the lower spectral gap of Cayley sum graphs.

Abstract: A $k$-regular partition into distinct parts is a partition into distinct parts with no part divisible by $k$. In this paper, we provide a general method to establish the unimodality of $k$-regular partition into distinct parts where the largest part is at most $km+k-1$. Let $d_{k,m}(n)$ denote the number of $k$-regular partition of $n$ into distinct parts where the largest part is at most $km+k-1$. In line with this method, we show that $d_{4,m}(n)\geq d_{4,m}(n-1)$ for $m\geq 0$, $1\leq n\leq 3(m+1)^2$ and $n\neq 4$ and $d_{8,m}(n)\geq d_{8,m}(n-1)$ for $m\geq 2$ and $1\leq n\leq 14(m+1)^2$. When $5\leq k\leq 10$ and $k\neq 8$, we show that $d_{k,m}(n)\geq d_{k,m}(n-1)$ for $m\geq 0$ and $1\leq n\leq \left\lfloor\frac{k(k-1)(m+1)^2}{4}\right\rfloor$.

Abstract: An $(r,z,k)$-mixed graph $G$ has every vertex with undirected degree $r$, directed in- and out-degree $z$, and diameter $k$. In this paper, we study the case $r=z=1$, proposing some new constructions of $(1,1,k)$-mixed graphs with a large number of vertices $N$. Our study is based on computer techniques for small values of $k$ and the use of graphs on alphabets for general $k$. In the former case, the constructions are either Cayley or lift graphs. In the latter case, some infinite families of $(1,1,k)$-mixed graphs are proposed with diameter of the order of $2\log_2 N$.

Abstract: We introduce, motivate and study $\varepsilon$-almost collision-flat (ACFU) universal hash functions $f:\mathcal X\times\mathcal S\to\mathcal A$. Their main property is that the number of collisions in any given value is bounded. Each $\varepsilon$-ACFU hash function is an $\varepsilon$-almost universal (AU) hash function, and every $\varepsilon$-almost strongly universal (ASU) hash function is an $\varepsilon$-ACFU hash function. We study how the size of the seed set $\mathcal S$ depends on $\varepsilon,|\mathcal X|$ and $|\mathcal A|$. Depending on how these parameters are interrelated, seed-minimizing ACFU hash functions are equivalent to mosaics of balanced incomplete block designs (BIBDs) or to duals of mosaics of quasi-symmetric block designs; in a third case, mosaics of transversal designs and nets yield seed-optimal ACFU hash functions, but a full characterization is missing. By either extending $\mathcal S$ or $\mathcal X$, it is possible to obtain an $\varepsilon$-ACFU hash function from an $\varepsilon$-AU hash function or an $\varepsilon$-ASU hash function, generalizing the construction of mosaics of designs from a given resolvable design (Gnilke, Greferath, Pav{\v c}evi\'c, Des. Codes Cryptogr. 86(1)). The concatenation of an ASU and an ACFU hash function again yields an ACFU hash function. Finally, we motivate ACFU hash functions by their applicability in privacy amplification.