Combinatorics
arXiv:math.CO
Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory.
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Trending in Combinatorics
2601.00766Set mappings for general graphs
The study of extremal problems for set mappings has a long history. It was introduced in 1958 by Erdős and Hajnal, who considered the case of cliques in graphs and hypergraphs. Recently, Caro, Patkós, Tuza and Vizer revisited this subject, and initiated the systematic study of set mapping problems for general graphs. In this paper, we prove the following result, which answers one of their questions. Let $G$ be a graph with $m$ edges and no isolated vertices and let $f : E(K_N) \rightarrow E(K_N)$ such that $f(e)$ is disjoint from $e$ for all $e \in E(K_N)$. Then for some absolute constant $C$, as long as $N \geq C m$, there is a copy $G^*$ of $G$ in $K_N$ such that $f(e)$ is disjoint from $V(G^*)$ for all $e \in E(G^*)$. The bound $N = O(m)$ is tight for cliques and is tight up to a logarithmic factor for all $G$.
2601.00766
Jan 2026Combinatorics
Tiling with Boundaries: Dense digital images have large connected components
If most of the pixels in an $n \times m$ digital image are the same color, must the image contain a large connected component? How densely can a given set of connected components pack in $\mathbb{Z}^2$ without touching? We answer these two closely related questions for both 4-connected and 8-connected components. In particular, we use structural arguments to upper bound the "white" pixel density of infinite images whose white (4- or 8-)connected components have size at most $k$. Explicit tilings show that these bounds are tight for at least half of all natural numbers $k$ in the 4-connected case, and for all $k$ in the 8-connected case. We also extend these results to finite images. To obtain the upper bounds, we define the exterior site perimeter of a connected component and then leverage geometric and topological properties of this set to partition images into nontrivial regions called polygonal tiles. Each polygonal tile contains a single white connected component and satisfies a certain maximality property. We then use isoperimetric inequalities to precisely bound the area of these tiles. The solutions to these problems represent new statistics on the connected component distribution of digital images.
2512.11740
Dec 2025Combinatorics
Fault-tolerant mutual-visibility: complexity and solutions for grid-like networks
Networks are often modeled using graphs, and within this setting we introduce the notion of $k$-fault-tolerant mutual visibility. Informally, a set of vertices $X \subseteq V(G)$ in a graph $G$ is a $k$-fault-tolerant mutual-visibility set ($k$-ftmv set) if any two vertices in $X$ are connected by a bundle of $k+1$ shortest paths such that: ($i$) each shortest path contains no other vertex of $X$, and ($ii$) these paths are internally disjoint. The cardinality of a largest $k$-ftmv set is denoted by $\mathrm{f}μ^{k}(G)$. The classical notion of mutual visibility corresponds to the case $k = 0$. This generalized concept is motivated by applications in communication networks, where agents located at vertices must communicate both efficiently (i.e., via shortest paths) and confidentially (i.e., without messages passing through the location of any other agent). The original notion of mutual visibility may fail in unreliable networks, where vertices or links can become unavailable. Several properties of $k$-ftmv sets are established, including a natural relationship between $\mathrm{f}μ^{k}(G)$ and $ω(G)$, as well as a characterization of graphs for which $\mathrm{f}μ^{k}(G)$ is large. It is shown that computing $\mathrm{f}μ^{k}(G)$ is NP-hard for any positive integer $k$, whether $k$ is fixed or not. Exact formulae for $\mathrm{f}μ^{k}(G)$ are derived for several specific graph topologies, including grid-like networks such as cylinders and tori, and for diameter-two networks defined by Hamming graphs and by the direct product of complete graphs.
2512.01978
Dec 2025Combinatorics
