Combinatorics (math.CO)

Abstract: We present a new explicit construction of onesided bipartite lossless expanders of constant degree, with arbitrary constant ratio between the sizes of the two vertex sets. Our construction is simpler to state and analyze than the prior construction of Capalbo, Reingold, Vadhan, and Wigderson (2002). We construct our lossless expanders by imposing the structure of a constant-sized lossless expander "gadget" within the neighborhoods of a large bipartite spectral expander; similar constructions were previously used to obtain the weaker notion of unique-neighbor expansion. Our analysis simply consists of elementary counting arguments and an application of the expander mixing lemma.

Abstract: An almost $k$-cover of the hypercube $Q^n = \{0,1\}^n$ is a collection of hyperplanes that avoids the origin and covers every other vertex at least $k$ times. When $k$ is large with respect to the dimension $n$, Clifton and Huang asymptotically determined the minimum possible size of an almost $k$-cover. Central to their proof was an extension of the LYM inequality, concerning a weighted count of hyperplanes. In this paper we completely characterise the hyperplanes of maximum weight, showing that there are $\binom{2n-1}{n}$ such planes. We further provide stability, bounding the weight of all hyperplanes that are not of maximum weight. These results allow us to effectively shrink the search space when using integer linear programming to construct small covers, and as a result we are able to determine the exact minimum size of an almost $k$-cover of $Q^6$ for most values of $k$. We further use the stability result to improve the Clifton--Huang lower bound for infinitely many choices of $k$ in every sufficiently large dimension $n$.

Abstract: In this paper, we have studied the distance magic labelling of Generalised Mycielskian of a few families of graphs.

Abstract: We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Moreover, two additional characterizations are expressed in terms of resonance digraphs and via local structures of inner duals of 2-connected outerplane bipartite graphs, respectively.

Abstract: In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under specific network structures can help scientists design and construct networks with high data transmission rates. A graph $G$ is called a fractional $(g,f)$-covered graph if for any $e\in E(G)$, $G$ admits a fractional $(g,f)$-factor covering $e$. A graph $G$ is called a fractional $(g,f,n)$-critical covered graph if after removing any $n$ vertices of $G$, the resulting graph of $G$ is a fractional $(g,f)$-covered graph. In this paper, we verify that if a graph $G$ of order $p$ satisfies $p\geq\frac{(a+b-1)(a+b-2)+(a+d)n+1}{a+d}$, $\delta(G)\geq\frac{(b-d-1)p+(a+d)n+a+b+1}{a+b-1}$ and $\delta(G)>\frac{(b-d-2)p+2\alpha(G)+(a+d)n+1}{a+b-2}$, then $G$ is a fractional $(g,f,n)$-critical covered graph, where $g,f:V(G)\rightarrow Z^{+}$ be two functions such that $a\leq g(x)\leq f(x)-d\leq b-d$ for all $x\in V(G)$, which is a generalization of Zhou's previous result [S. Zhou, Some new sufficient conditions for graphs to have fractional $k$-factors, International Journal of Computer Mathematics 88(3)(2011)484--490].

Abstract: By prepending zeros to a given sequence Hankel determinants of backward shifts of this sequence become meaningful. We obtain some results for the sequences of Catalan numbers and of some numbers and polynomials which are related to Catalan numbers and propose conjectures for sequences of convolution powers of Catalan numbers.

Abstract: Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of vertex degrees of $G$. For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$. The largest eigenvalue of $A_\alpha(G)$ is called the $\alpha$-index or the $A_\alpha$-spectral radius of $G$. A graph is minimally $k$-(edge)-connected if it is $k$-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not $k$-(edge)-connected. In this paper, we characterize the minimally 2-edge-connected graphs and minimally 3-connected graph with given order having the maximum $\alpha$-index for $\alpha \in [\frac{1}{2},1)$, respectively.

Abstract: In this paper we prove that $\mathbb{P}(|X| \geq \sqrt{\text{Var}(X)}) \geq 7/32$ for every finite Rademacher sum $X$, confirming a conjecture by Hitczenko and Kwapie{\'n} from 1994, and improving upon results from Burkholder, Oleszkiewicz, and Dvo\v{r}\'ak and Klein. Moreover we fully determine the function $f(y)= \inf_X \mathbb{P}(|X| \geq y\sqrt{\text{Var}(X)})$ where the $\inf$ is taken over all finite Rademacher sums $X$, confirming a conjecture by Lowther and giving a partial answer to a question by Keller and Klein.

Abstract: Given a group $G$ and a subgroup $H \leq G$, a set $\mathcal{F}\subset G$ is called $H$\emph{-intersecting} if for any $g,g' \in \mathcal{F}$, there exists $xH \in G/H$ such that $gxH=g'xH$. The \emph{intersection density} of the action of $G$ on $G/H$ by (left) multiplication is the rational number $\rho(G,H)$, equal to the maximum ratio $\frac{|\mathcal{F}|}{|H|}$, where $\mathcal{F} \subset G$ runs through all $H$-intersecting sets of $G$. The \emph{intersection spectrum} of the group $G$ is then defined to be the set $$ \sigma(G) := \left\{ \rho(G,H) : H\leq G \right\}. $$ It was shown by Bardestani and Mallahi-Karai [{\it J. Algebraic Combin.}, 42(1):111-128, 2015] that if $\sigma(G) = \{1\}$, then $G$ is necessarily solvable. The natural question that arises is, therefore, which rational numbers larger than $1$ belong to $\sigma(G)$, whenever $G$ is non-solvable. In this paper, we study the intersection spectrum of the linear group $\operatorname{PSL}_2(q)$. It is shown that $2 \in \sigma\left(\operatorname{PSL}_2(q)\right)$, for any prime power $q\equiv 3 \pmod 4$. Moreover, when $q\equiv 1 \pmod 4$, it is proved that $\rho(\operatorname{PSL}_2(q),H)=1$, for any odd index subgroup $H$ (containing $\mathbb{F}_q$) of the Borel subgroup (isomorphic to $\mathbb{F}_q\rtimes \mathbb{Z}_{\frac{q-1}{2}}$) consisting of all upper triangular matrices.

Abstract: The numbers we study in this paper are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that contain the word $p$ (as a subsequence) exactly $k$ times. Our motivation comes from the analogous study of pattern containment in permutations. In our first set of results, we obtain explicit expressions for $B_{n, p}(k)$ for small values of $k$. We then focus on words $p$ with at most $3$ runs and study the maximum number of occurrences of $p$ a word of length $n$ can have. We also study the internal zeros in the sequence $(B_{n, p}(k))_{k \geq 0}$ for fixed $n$ and discuss the unimodality and log-concavity of such sequences.

Abstract: Bassino et al. (arXiv:1907.08517) have shown that uniform random co-graphs (graphs without induced $P_4$) of size $n$ converge to a certain non-deterministic graphon. The edge-density of this graphon is a random variable $\Lambda \in [0,1]$ whose first moments have been computed by these authors. The first purpose of this note is to observe that, in fact, these moments can be computed by a simple recurrence relation. The problem leads us to the following question of independent interest: given $k$ i.i.d. uniform pairs of points $(X_1,Y_1),$ $\dots$, $(X_k,Y_k)$ in the Brownian CRT, what is the size $S_k$ of the set $\{X_i\wedge Y_i, i=1\dots k\}$ formed by their pairwise last common ancestors? We show that ${S_k}\sim c{\sqrt{k}\log k}$ in probability, with $c=(\sqrt{2\pi})^{-1}$. The method to establish the recurrence relation is reminiscent of Janson's computation of moments of the Wiener index of (large) random trees. The logarithm factor in the convergence result comes from the estimation of Riemann sums in which summands are weighted by the integer divisor function -- such sums naturally occur in the problem. We have not been able to analyse the asymptotics of moments of $\Lambda$ directly from the recurrence relation, and in fact our study of $S_k$ is independent from it. Several things remain to be done, in particular we only scratch the question of large deviations of $S_k$, and the precise asymptotics of moments of $\Lambda$ is left open.

Abstract: Graph theory is an interdisciplinary field of study that has various applications in mathematical modeling and computer science. Research in graph theory depends on the creation of not only theorems but also conjectures. Conjecture-refuting algorithms attempt to refute conjectures by searching for counterexamples to those conjectures, often by maximizing certain score functions on graphs. This study proposes a novel conjecture-refuting algorithm, referred to as the adaptive Monte Carlo search (AMCS) algorithm, obtained by modifying the Monte Carlo tree search algorithm. Evaluated based on its success in finding counterexamples to several graph theory conjectures, AMCS outperforms existing conjecture-refuting algorithms. The algorithm is further utilized to refute six open conjectures, two of which were chemical graph theory conjectures formulated by Liu et al. in 2021 and four of which were formulated by the AutoGraphiX computer system in 2006. Finally, four of the open conjectures are strongly refuted by generalizing the counterexamples obtained by AMCS to produce a family of counterexamples. It is expected that the algorithm can help researchers test graph-theoretic conjectures more effectively.