Combinatorics (math.CO)

Abstract: We give a more coherent and apparent definition of upward planar order.

Abstract: We prove that the $k$-power of any planar graph $G$ is contained in $H\boxtimes P\boxtimes K_{f(\Delta(G),k)}$ for some graph $H$ with treewidth $15\,288\,899$, some path $P$, and some function $f$. This resolves an open problem of Ossona de Mendez. In fact, we prove a more general result in terms of shallow minors that implies similar results for many `beyond planar' graph classes, without dependence on $\Delta(G)$. For example, we prove that every $k$-planar graph is contained in $H\boxtimes P\boxtimes K_{f(k)}$ for some graph $H$ with treewidth $15\,288\,899$ and some path $P$, and some function $f$. This resolves an open problem of Dujmovi\'c, Morin and Wood. We generalise all these results for graphs of bounded Euler genus, still with an absolute bound on the treewidth. At the heart of our proof is the following new concept of independent interest. An $\ell$-blocking partition of a graph $G$ is a partition of $V(G)$ into connected sets such that every path of length greater than $\ell$ in $G$ contains at least two vertices in one part. We prove that every graph of Euler genus $g$ has a $894$-blocking partition with parts of size bounded by a function of $\Delta(G)$ and $g$. Motivated by this result, we study blocking partitions in their own right. We show that every graph has a $2$-blocking partition with parts of size bounded by a function of $\Delta(G)$ and $\textrm{tw}(G)$. On the other hand, we show that 4-regular graphs do not have $\ell$-blocking partitions with bounded size parts.

Abstract: The minimum and maximum cuts of an undirected edge-weighted graph are classic problems in graph theory. While the Min-Cut Problem can be solved in P, the Max-Cut Problem is NP-Complete. Exact and heuristic methods have been developed for solving them. For both problems, we introduce a natural extension in which cutting an edge induces a cost. Our goal is to find a cut that minimizes the sum of the cut weights but, at the same time, restricts its total cut cost to a given budget. We prove that both restricted problems are NPComplete and we also study some of its properties. Finally, we develop exact algorithms to solve both as well as a non-exact algorithm for the min-cut case based on a Lagreangean relaxation that generally provides optimal solutions. Their performance is reported by an extensive computational experience.

Abstract: Let $d \ge 3$ be an integer. We show that whenever an order-$d$ tensor admits $d+1$ decompositions according to Tao's slice rank, if the linear subspaces spanned by their one-variable functions constitute a sunflower for each choice of special coordinate, then the tensor admits a decomposition where these linear subspaces are contained in the centers of these respective sunflowers. As an application, we deduce that for every nonnegative integer $k$ and every finite field $\mathbb{F}$ there exists an integer $C(d,k,|\mathbb{F}|)$ such that every order-$d$ tensor with slice rank $k$ over $\mathbb{F}$ admits at most $C(d,k,|\mathbb{F}|)$ decompositions with length $k$, up to a class of transformations that can be easily described.

Abstract: Pachner proved that all closed combinatorially equivalent combinatorial manifolds can be transformed into each other by a finite sequence of bistellar moves. We prove an analogue of Pachner's theorem for combinatorial manifolds with a free Z2-action, and use it to give a combinatorial proof of Fan's lemma about labellings of centrally symmetric triangulations of spheres. Similarly to other combinatorial proofs, we must assume an additional property of the triangulation for the proof to work. However, unlike the other combinatorial proofs, no such assumption is needed for dimensions at most 3.

Abstract: Let $G$ be a graph, $H$ be a subgraph of $G$, and let $G- H$ be the graph obtained from $G$ by removing a copy of $H$. Let $K_{1, n}$ be the star on $n+ 1$ vertices. Let $t\geq 2$ be an integer and $H_{1}, \dots, H_{t}$ and $H$ be graphs, and let $H\rightarrow (H_{1}, \dots, H_{t})$ denote that every $t$ coloring of $E(H)$ yields a monochromatic copy of $H_{i}$ in color $i$ for some $i\in [t]$. Ramsey number $r(H_{1}, \dots, H_{t})$ is the minimum integer $N$ such that $K_{N}\rightarrow (H_{1}, \dots, H_{t})$. Star-critical Ramsey number $r_{*}(H_{1}, \dots, H_{t})$ is the minimum integer $k$ such that $K_{N}- K_{1, N- 1- k}\rightarrow (H_{1}, \dots, H_{t})$ where $N= r(H_{1}, \dots, H_{t})$. Let $rr(H_{1}, \dots, H_{t})$ be the regular Ramsey number for $H_{1}, \dots, H_{t}$, which is the minimum integer $r$ such that if $G$ is an $r$-regular graph on $r(H_{1}, \dots, H_{t})$ vertices, then $G\rightarrow (H_{1}, \dots, H_{t})$. Let $m_{1}, \dots, m_{t}$ be integers larger than one, exactly $k$ of which are even. In this paper, we prove that if $k\geq 2$ is even, then $r_{*}(K_{1, m_{1}}, \dots, K_{1, m_{t}})= \sum_{i= 1}^{t} m_{i}- t+ 1- \frac{k}{2}$ which disproves a conjecture of Budden and DeJonge in 2022. Furthermore, we prove that if $k\geq 2$ is even, then $rr(K_{1, m_{1}}, \dots, K_{1, m_{t}})= \sum_{i= 1}^{t} m_{i}- t$. Otherwise, $rr(K_{1, m_{1}}, \dots, K_{1, m_{t}})= \sum_{i= 1}^{t} m_{i}- t+ 1$.

Abstract: We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell\leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least $$\cases{\left(\dfrac{1}{2}+o(1)\right)n & if $(k-\ell)\mid k$,\\ & \\ \left(\dfrac{1}{\lceil \frac{k}{k-\ell}\rceil(k-\ell)}+o(1)\right)n & if $(k-\ell)\nmid k$,}$$ contains $\exp(n\log n-\Theta(n))$ Hamilton $\ell$-cycles as long as $(k-\ell)\mid n$. When $(k-\ell)\mid k$ this gives a simple proof of a result of Glock, Gould, Joos, K\"uhn and Osthus, while, when $(k-\ell)\nmid k$ this gives a weaker count than that given by Ferber, Hardiman and Mond or, when $\ell<k/2$, by Ferber, Krivelevich and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.

Abstract: Strengthening the classical concept of Steiner trees, West and Wu [J. Combin. Theory Ser. B 102 (2012), 186--205] introduced the notion of a $T$-connector in a graph $G$ with a set $T$ of terminals. They conjectured that if the set $T$ is $3k$-edge-connected in $G$, then $G$ contains $k$ edge-disjoint $T$-connectors. We disprove this conjecture by constructing infinitely many counterexamples for $k=1$ and for each even $k$.

Abstract: We study the action of Bender-Knuth involutions on linear extensions of posets and identify LE-cactus posets, i.e. those for which the cactus relations hold. It was conjectured in \cite{chiang2023bender} that d-complete posets are LE-cactus. Among the non-d-complete posets that are LE-cactus, one notable family is ordinal sums of antichains. In this paper, we characterize the LE-cactus posets in a more general family, namely ordinal sums of disjoint unions of chains.

Abstract: Webs yield an especially important realization of certain Specht modules, irreducible representations of symmetric groups, as they provide a pictorial basis with a convenient diagrammatic calculus. In recent work, the last three authors associated polynomials to noncrossing partitions without singleton blocks, so that the corresponding polynomials form a web basis of the pennant Specht module $S^{(d,d,1^{n-2d})}$. These polynomials were interpreted as global sections of a line bundle on a 2-step partial flag variety. Here, we both simplify and extend this construction. On the one hand, we show that these polynomials can alternatively be situated in the homogeneous coordinate ring of a Grassmannian, instead of a 2-step partial flag variety, and can be realized as tensor invariants of classical (but highly nonplanar) tensor diagrams. On the other hand, we extend these ideas from the pennant Specht module $S^{(d,d,1^{n-2d})}$ to more general flamingo Specht modules $S^{(d^r,1^{n-rd})}$. In the hook case $r=1$, we obtain a spanning set that can be restricted to a basis in various ways. In the case $r>2$, we obtain a basis of a well-behaved subspace of $S^{(d^r,1^{n-rd})}$, but not of the entire module.

Abstract: Let $G$ be a connected graph with $|V| = n$ and $|E| = m$. A bijection $f:E\rightarrow \{1,2,...,m\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, $w(u) \neq w(v)$, where $w(u) = \sum_{e \in E(u)}f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ where the vertex $v$ is assigned the color $w(v)$. The local antimagic chromatic number is the minimum number of colors taken over all colorings induced by local antimagic labelings of $G$. Let $m,n > 1$. In this paper, the local antimagic chromatic number of a complete tripartite graph $K_{1,m,n}$, and $r$ copies of a complete bipartite graph $K_{m,n}$ where $m \not \equiv n \bmod 2$ are determined.