Combinatorics (math.CO)

Abstract: In [Distrance-regular Cayley graphs on dihedral groups, J. Combin. Theory Ser B 97 (2007) 14--33], Miklavi\v{c} and Poto\v{c}nik proposed the problem of characterizing distance-regular Cayley graphs, which can be viewed as an extension of the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, all distance-regular Cayley graphs over $\mathbb{Z}_{p^s}\oplus\mathbb{Z}_{p}$ with $p$ being an odd prime are determined. It is shown that every such graph is isomorphic to a complete graph, a complete multipartite graph, or the line graph of a transversal design $TD(r,p)$ with $2\leq r\leq p-1$.

Abstract: For any polyhedral norm, the bisector of two points is a polyhedral complex. We study combinatorial aspects of this complex. We investigate the sensitivity of the presence of labelled maximal cells in the bisector relative to the position of the two points. We thereby extend work of Criado, Joswig and Santos (2022) who showed that for the tropical distance function the presence of maximal cells is encoded by a polyhedral fan, the bisection fan. We initiate the study of bisection cones and bisection fans with respect to arbitrary polyhedral norms. In particular, we show that the bisection fan always exists for polyhedral norms in two dimensions. Furthermore, we determine the bisection fan of the $\ell_{1}$-norm and the $\ell_{\infty}$-norm as well as the discrete Wasserstein distance in arbitrary dimensions. Intricate combinatorial structures, such as the resonance arrangement, make their appearance. We apply our results to obtain bounds on the combinatorial complexity of the bisectors.

Abstract: Let $G$ be a graph and $X\subseteq V(G)$. Then, vertices $x$ and $y$ of $G$ are $X$-visible if there exists a shortest $u,v$-path where no internal vertices belong to $X$. The set $X$ is a mutual-visibility set of $G$ if every two vertices of $X$ are $X$-visible, while $X$ is a total mutual-visibility set if any two vertices from $V(G)$ are $X$-visible. The cardinality of a largest mutual-visibility set (resp. total mutual-visibility set) is the mutual-visibility number (resp. total mutual-visibility number) $\mu(G)$ (resp. $\mu_t(G)$) of $G$. It is known that computing $\mu(G)$ is an NP-complete problem, as well as $\mu_t(G)$. In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, cube-connected cycles, and butterflies). Concerning computing $\mu(G)$, we provide approximation algorithms for both hypercubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing $\mu_t(G)$ (in the literature, already studied in hypercubes), we provide exact formulae for both cube-connected cycles and butterflies.

Abstract: We show that cylindric partitions are in one-to-one correspondence with a pair which has an ordinary partition and a colored partition into distinct parts. Then, we show the general form of the generating function for cylindric partitions into distinct parts and give some examples. We prove part of a conjecture by Corteel, Dousse, and Uncu. With due computational support; the other part of Corteel, Dousse, and Uncu's conjecture, which also appeared in Warnaar's work, is extended. The approaches and proofs are elementary and combinatorial.

Abstract: Let ${\rm Mat}_n(\mathbb{F})$ denote the set of square $n\times n$ matrices over a field $\mathbb{F}$ of characteristic different from two. The permanental rank ${\rm prk}\,(A)$ of a matrix $A \in{\rm Mat}_{n}(\mathbb{F})$ is the size of the maximal square submatrix in $A$ with nonzero permanent. By $\Lambda^{k}$ and $\Lambda^{\leq k}$ we denote the subsets of matrices $A \in {\rm Mat}_{n}(\mathbb{F})$ with ${\rm prk}\,(A) = k$ and ${\rm prk}\,(A) \leq k$, respectively. In this paper for each $1 \leq k \leq n-1$ we obtain a complete characterization of linear maps $T: {\rm Mat}_{n}(\mathbb{F}) \to {\rm Mat}_{n}(\mathbb{F})$ satisfying $T(\Lambda^{\leq k}) = \Lambda^{\leq k}$ or bijective linear maps satisfying $T(\Lambda^{\leq k}) \subseteq \Lambda^{\leq k}$. Moreover, we show that if $\mathbb{F}$ is an infinite field, then $\Lambda^{k}$ is Zariski dense in $\Lambda^{\leq k}$ and apply this to describe such bijective linear maps satisfying $T(\Lambda^{k}) \subseteq \Lambda^{k}$.

Abstract: A closed form is found for the expected number of rolls of a fair n-sided die until three consecutive increasing values are seen. The answer is rational, and the greatest common divisor of the numerator and denominator is given in terms of n. As n goes to infinity, the probability generating function is found for the limiting case, which is also the exponential generating function for permutations ending in a double rise and without other double rises. Thus exact values are found for the limiting expectation and variance, which are approximately 7.92437 and 27.98133 respectively.

Abstract: The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory. We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of (Bonnet and D\'epr\'es 2022), which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain an optimal linear bound on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.