Papers

We prove that every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes (products of two distinct primes). This is the $\omega=2$ integer case of a problem of Erdős and Graham, stated only as a conjecture by Butler, Erdős and Graham (Integers 15 (2015), A51), who proved the $\omega=3$ analogue. Counterintuitively the problem hardens as $\omega$ decreases – the induction's feed thins – so $\omega=2$ is the hard case; our proof adapts the Butler-Erdős-Graham induction to this thin-feed regime, where the entire content of the induction step reduces to an explicit onset inequality $Y_0(N)\le\min\{\beta(N),\beta'(N)\}$, proved for all $N\ge10$ by Olson's addition theorem and elementary Chebyshev bounds above a finite, machine-checked base range. The same engine extends to the rationals: for every squarefree $b$, every $a/b$ above an explicit threshold $\min\{B_{N_b}/6,\,1/5\}$ is $\omega=2$ representable, unconditionally. As an application we give the first complete proof of the rational $\omega=3$ statement – every $a/b$ with squarefree $b$ is a sum of distinct sphenic unit fractions – that Butler, Erdős and Graham conjectured but left unpublished; a descent settles every $\omega\ge3$. What remains open is the $\omega=2$ regime below this threshold, which we reduce to a single explicit conjecture – that the gap-free floor of a semiprime subset-sum set tends to zero. This work is a human-AI collaboration: AI tools (notably Anthropic's Claude, used through Claude Code) contributed substantially to the Lean formalisation, the experiments, and the writing; correspondingly, every result is machine-checked in Lean 4 / Mathlib (no sorry; two cited classical axioms, plus the native_decide compiler-trust base for the finite computations), so its correctness is independent of the tools used.

We develop a uniform skew Murnaghan–Nakayama theory for graded Hopf dual pairs equipped with a nondegenerate Hopf pairing. Using the completed Cauchy element, its grouplike factorization, and the resulting partial contraction operators, we establish a general skew Cauchy identity together with an abstract skew Murnaghan–Nakayama rule. Specializing this framework recovers and extends the classical skew Murnaghan–Nakayama rule for symmetric functions, and yields new skew Murnaghan–Nakayama formulas in several settings, including the dual pairs $(\mathrm{NSym}, \mathrm{QSym}) $ and $(\Lambda^{(k)}, \Lambda_{(k)}) $ arising in $k$-Schur theory, as well as the type $C$ affine Grassmannian context. As applications, we obtain generating functions for irreducible characters of Ariki–Koike algebras, including their type $A$ and type $B$ specializations, as well as Hecke–Clifford algebras and $\mathfrak q$-rook monoid algebras. We also give ribbon-tableau expansions for skew $(q,t)$-Kostka polynomials and for the entries of the inverse transition matrix, thereby answering a question of Carbonara (1998). Finally, by specializing the auxiliary alphabet $Y$ to sums of powers of primitive roots of unity, we derive a skew plethystic Murnaghan–Nakayama formula together with a Schur expansion for skew modular Schur functions; as a further consequence, we confirm Walker's conjecture (1994) by showing that if the transition from the modular Schur functions to the Schur basis is trivial in the row indexed by $\lambda$, then $\lambda$ must be a $k$-core.

We solve the inverse problem for Ising models on reduced planar graphs in a disk, i.e., recovering the edge coupling constants from the boundary spin correlations. A recursive solution to this problem was provided by Galashin–Pylyavskyy. Our solution is non-recursive. It is based on an Ising analog of the chamber ansatz which asserts that the inverse map should factor through variables on the graph that transform under the Ising Y-$\Delta$ move according to the discrete CKP equation.

The Constructor–Blocker game is played on the edge set of $K_n$. Two players alternately claim previously unclaimed edges. Constructor aims to maximize the number of copies of a target graph $H$ in her graph while keeping it $F$-free throughout the game, whereas Blocker aims to minimize this number. When both players play optimally and Constructor moves first, the final number of copies of $H$ in Constructor's graph is called the score of the game and is denoted by $g(n,H,F)$. Recently, Balogh, Chen, and English systematically studied this game for non-bipartite forbidden graphs $F$. However, bipartite forbidden graphs present additional difficulties. In this paper, we focus on the case in which the forbidden graph is an even cycle. First, using a finite-field-geometric construction, we confirm a conjecture of Balogh, Chen and English by proving \[ g(n,K_3,C_4)=\Theta(n^{3/2}). \] We also establish two-sided bounds for $g(n,K_3,C_{2k})$ that relate the game score to classical extremal numbers. Our proofs reveal a structural distinction between the cases $k=2$ and $k\ge3$: a vertex-duplication method works for longer even cycles, but it necessarily creates $4$-cycles and therefore cannot be used in the $C_4$-free game. Finally, we determine the exact order of $g(n,C_t,C_{2k})$ for all $k\ge2$ and all $t\ge4$.

Let $G$ and $H$ be connected graphs that are 2-isomorphic. It is known that their Laplacian matrices are congruent by a unimodular matrix $U$. In this paper we show (Thm. \ref{thm:main2}) that $U$ is a bijection between certain spaces of harmonic vectors on the vertices of $G$ and $H$. In particular (Cor. \ref{cor:main1}) if $u$ is a harmonic vector with respect to vertices $c, d$ in $H$ and the 2-isomorphism maps edge $(a,b)$ in $G$ to edge $(c,d)$ in $H$, then $uU$ is a harmonic vector with respect to vertices $a, b$ in $G$.

The problems of designing sparse networks arise frequently in resource allocation and operations research. In production systems, for example, sparse process flexibility designs are used to handle uncertain demand effectively: the goal is to construct the sparsest bipartite graph between supply and demand that still achieves an expected fulfilled demand comparable to that of a fully flexible system. In middle-mile transportation, sparse delivery-route subgraphs that sustain large matchings after random node deletions help reduce delivery costs; here, the goal is to design the sparsest graph whose maximum matching size remains comparable to that of the fully connected graph under node deletions. The design of sparse networks has been studied extensively, with state-of-the-art results providing order-wise optimal designs for both bipartite and unipartite networks (Chen et al., 2015; Feng et al., 2024). However, identifying designs that achieve the sharp theoretical limit – where the average degree asymptotically matches the lower bound of any graph to achieve a given loss level, has remained open. In this paper, we prove that the random regular graph achieves this sharp optimal condition in both bipartite and unipartite settings. Numerical experiments further validate this optimality. Our results highlight a practical guideline for sparse flexibility networks: designs that combine degree regularity with dispersed edge placement can achieve optimal performance under uncertainty.

It is a classical theorem of Robertson and Seymour (1986) that the treewidth of a graph is linearly related to its separation number: the smallest integer $k$ such that, for every weight function on the vertices, the graph admits a balanced separator of size at most $k$. Motivated by recent progress on coarse treewidth, Abrishami, Czyżewska, Kluk, Pilipczuk, Pilipczuk, and Rzażewski (2025) conjectured the following coarse analogue: for every $r\in \mathbb{N}$ there exists an $r'\in \mathbb{N}$ such that every graph that admits balanced separators that can be covered by a bounded number of balls of bounded radius $r$ admits a tree decomposition where every bag can be covered by a bounded number of balls of radius $r'$. We verify a stronger variant of this conjecture for all $r \in \mathbb{N}$ for the hereditary class of $K_{t,t}$-induced-minor-free graphs of bounded clique number. A key step in the proof is the following result, which we expect to be of independent interest. In $K_{t,t}$-induced-minor-free graphs with clique number bounded by $s$, given a large subset of vertices $Y \subseteq V(G)$, there is a set $Z$ whose size is bounded by a function polynomial in $s$, such that no ball of radius $r$ in $G- Z$ covers a large proportion of $Y$.

The valley Delta conjecture of Haglund, Remmel and Wilson predicts that the symmetric function $\Delta'_{e_{n-k-1}} e_n$ equals a generating function over labelled Dyck paths with $k$ decorated contractible valleys. Unlike the rise version, which is now a theorem, the valley version remains open; indeed it is not even known that its combinatorial side is symmetric. We prove that the coefficients of $t^0$ and $t^1$ in the valley generating function are symmetric functions for all $n\ge 1$ and $k\ge 0$; equivalently, the fixed-diagonal-multiset slices of area at most one are symmetric. The theorem follows from an adjacent exchange identity for scaffold classes, which we prove in a strictly stronger form refined by the numbers of undecorated rows carrying the labels $r,r+1$ between consecutive rows with other labels. The proof develops a transfer-operator calculus in a $q$-deformed two-variable algebra generated by two commuting half-twists. In this algebra the exchange reduces to two scalar symmetric-series identities for the operator $T=u\mathfrak{d}+v$. We also verify the refined identity computationally at area two over extensive finite ranges and state the resulting general conjecture.

In this paper, we obtain several Ramanujan-type congruences modulo 5 and 7 for sum of certain non-overlined parts in overpartitions classified by parity and sum of certain parts in partitions without repeated odd parts classified by parity. Our proofs for the congruences are elementary, depending only on classical theta function identities.

We give a brief exposition of a result of Song, Nguyen, and Lin (2026) that every election (with ranked preferences) has a Condorcet winning set of at most five candidates.

Jackson and Tanigawa conjectured that the rigidity matroid $\mathcal{R}^d_n$ and the hyperconnectivity matroid $\mathcal{H}^d_n$ are the unique maximal matroids in the posets of $\{K_{d+2}, K_{d+2,d+2}\}$-matroids and $\{K_{d+2},K_{d+1,d+1}\}$-matroids, respectively. We disprove these conjectures by showing the existence of maximal matroids that are distinct from the proposed candidates.

Pairwise compatibility graphs (PCGs) represent graph adjacency by an interval of leaf-to-leaf distances in a weighted tree. We study grid graphs under the PCG model and two natural extensions: multi-interval PCGs and OR-PCGs. First, we prove that every $d$-dimensional grid graph is a $(d-1)$-interval-PCG. The construction decomposes the grid into hyperplanes of constant coordinate sum and uses a large-base encoding so that distances between consecutive hyperplanes identify the coordinate direction of an edge. A pair of nearby code values is then merged into one interval, reducing the number of intervals from $d$ to $d-1$. Second, we prove that every $d$-dimensional grid is a $\lceil d/2\rceil$-OR-PCG by grouping coordinate directions into pairs; each paired-direction graph is a disjoint union of two-dimensional grid graphs and is therefore a PCG. Finally, an exact tree-metric satisfiability computation shows that $P_3\square P_3\square P_3$ is not a PCG. Consequently, the minimum number of intervals sufficient for all three-dimensional grid graphs is exactly two, resolving a previously posed open problem. The same obstruction shows that the OR-PCG bound is tight in dimension three and implies that every grid with at least three factors of order at least three is not a PCG.

Problem C7 from the 2006 IMO Shortlist gives \(A(P)-B(P)=V(P)-1\) for a generic convex polyhedron, where \(A(P)\) counts antipodal vertex pairs and \(B(P)\) counts antipodal edge-midpoint pairs. For an arbitrary convex polyhedron \(P\subset\mathbb{R}^3\), define \(\delta(P)=V(P)-1-A(P)+B(P)\). We construct a two-dimensional antipodal square complex \(X(P)\) and prove \(H_0(X(P);\mathbb{Z})\cong\mathbb{Z}\), \(H_1(X(P);\mathbb{Z})\cong\mathbb{Z}/2\), and \(H_2(X(P);\mathbb{Z})\cong\mathbb{Z}^{\delta(P)}\). Consequently, \(\delta(P)\geq 0\), extending the generic identity to the inequality \(A(P)-B(P)\leq V(P)-1\). The proof uses a directed double cover, a polyhedral support blow-up over the normal sphere, and the Vietoris–Begle mapping theorem. Independently, Euler integration on the projective normal fan gives an exact local formula for the defect; in three dimensions, only exact edge–facet and facet–facet opposite pairs contribute. We also determine the integral image of the square-boundary map: it is the even-cycle lattice of the antipodal graph, with nonzero Smith invariant factors \(1,\ldots,1,2\). Applications include a zero-defect criterion, centrally symmetric and extremal formulas, and an explicit description of defects and primitive belts for pyramids over polygons.

Let $G_{n,p}$ be the binomial random graph of density $p$ and let $X_H$ be the number of copies of a fixed graph $H$ in $G_{n,p}$. We prove asymptotically tight bounds on the logarithmic upper-tail probability of $X_H$ whenever $H$ is a connected, irregular graph with maximum degree $\Delta \ge 2$ and $p \ge n^{-1/\Delta - \varepsilon_H} (\log n)^{\omega(1)}$ for an explicit $\varepsilon_H >0$. These bounds are expressed in terms of a new variational problem that generalises the combinatorial optimisation problem arising from the naïve mean-field approximation. This new variational problem includes an entropy term that corresponds to the large number of embeddings of certain highly structured graphs in $K_n$. For a certain class of irregular graphs $H$ that we call stable, we show that this description of the upper-tail probability is valid in a range of densities that is optimal up to a poly($\log\log n$) factor. For a further subclass of stable graphs, which includes all irregular complete bipartite graphs, we show that this range of densities is optimal up to a multiplicative constant.

The Laplacian spectral radius of a graph is the largest eigenvalue of its Laplacian matrix. Previously, upper bounds for the Laplacian spectral radius were proposed using a backward-reconstruction procedure starting from expressions equal to $2x$ and substituting local degree data. A numbered list of 68 such candidate bounds was subsequently investigated, resulting in the refutation of 30 of these bounds; two additional bounds were later refuted in a separate study. This paper updates the status of the remaining 36 candidate bounds. Of these remaining bounds, we confirm 22 and refute 12, leaving only two upper bounds open. The valid bounds follow primarily from classical Laplacian spectral radius bounds and the Collatz–Wielandt comparison; the refutations are carried out through explicit counterexamples relying on equitable partitions.

This paper resolves several open questions regarding the algebraic and metric inheritance of distance-based regularities under graph products, with a particular focus on Commutative Distance Degree-Regular (CDDR) graphs. We characterize the behavior of CDDR, distance mean-regular (DMR), and distance degree-regular (DDR) graphs under the strong and direct (tensor) products. To address the non-closure of classical regularities under the direct product, we introduce and study a novel, broader family of graphs: Distance Walk Regular (DWR) graphs. Furthermore, we relax the global commutativity condition by defining and structuralizing the class of $i$-CDDR graphs, providing deep characterizations within the realm of Cayley graphs. By developing systematic lifting operations, we construct infinite families of graphs with arbitrarily large order and diameter within specific structural intersections, effectively settling topological gaps in the current regularity hierarchy.

In this note, we show that, for all $n\ge 2$, the number of distinct rooted binary phylogenetic $X$-trees displayed by a binary tree-child network $\mathcal{N}$ on $X$ with $n$ leaves is at most $2^{n-1}-1$ and that this upper bound is sharp. Furthermore, if $\mathcal{N}$ displays exactly $2^{n-1}-1$ such trees, then exactly one rooted binary phylogenetic $X$-tree is displayed twice, and this tree can be canonically found by iteratively replacing a reticulated cherry with a cherry.

Inclusion minimal convex bodies $K$ with the property that the integer translates of $K$ cover the space are studied. Such bodies are referred to as minimal covering bodies and it is shown that, while they are not necessarily tiles, they are polytopes with at least $2d$ facets, if $d$ is the dimension of $K$. Moreover, minimal covering bodies are related to covering properties of Minkowski combinations of convex bodies. Two sharp Brunn Minkowski type inequalities are established for the covering radius of the Minkowski sum of planar convex bodies.

Vizing's conjecture from 1963, considered by many the most important open problem in the field of graph domination, states that all graphs $G$ and $H$ satisfy $$\gamma(G\square H)\ge \gamma(G)\gamma(H),$$ where $\gamma$ denotes the domination number and $\square$ the Cartesian product. In a seminal result, Clark and Suen (2000) proved an approximate form of the conjecture, namely that $\gamma(G\square H)\ge \frac{1}{2}\gamma(G)\gamma(H)$ for all graphs $G$ and $H$. Despite several lower-order improvements of this bound and improvements for special classes of graphs $G$ and $H$, no absolute constant $c>\frac{1}{2}$ such that $\gamma(G\square H)\ge c\gamma(G)\gamma(H)$ for all graphs $G$ and $H$, has been known thus far. In this paper, we obtain the first constant-factor improvement of the Clark-Suen bound by proving that for all graphs $G$ and $H$, we have $$\gamma(G\square H)\ge c\gamma(G)\gamma(H),$$ where $$c=\frac{5+\sqrt{73}}{24}\approx 0.5643.$$ Along the way, we prove another lower bound on $\gamma(G\square H)$ which outperforms the above bound for many graphs.

Recently, the Turán problem for graphs with bounded clique number has attracted considerable attention. Since a graph can be regarded as a signed graph without negative edges, it is natural to extend the study of such Turán-type problems to signed graphs. With this motivation, we investigate the Turán-type problem for unbalanced $C_{2k+1}^{-}$-free signed graphs with bounded clique number. In fact, we establish a general result via the classical stability theorem. Specifically, for sufficiently large $n$ and a color-critical graph $F$, we determine the maximum number of edges among all $n$-vertex $C_{2k+1}^{-}$-free unbalanced signed graphs whose underlying graphs are $F$-free.

The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involution and a bijection that establish distributional equivalences for two classes of length-$2$ mesh patterns, thereby resolving a conjecture from 2019 and a recent conjecture. As a consequence, the best known upper bounds for the numbers of distribution-equivalence and Wilf-equivalence classes drop to $106$ and $47$, respectively. Combined with the known lower bounds of 105 and 46, conjectured to be exact, these results leave both classifications hinging on a single distribution-equivalence question conjectured in 2019, whose resolution would at once settle the remaining Wilf-equivalence case. We further conjecture that this unresolved equidistribution also holds for involutions, a subclass of all permutations. We also determine the distributions of three additional classes of length-$2$ mesh patterns through a detailed structural analysis. Our work combines bijective techniques with generating-function methods, yielding new insights into the structure and enumeration of short mesh patterns.

André planes of order $q^t$ can be constructed from the Desarguesian plane $PG(2,q^t)$ using a process that involves replacing André nets. This article constructs ovals in André planes of order $q^t$, $q$ even and $t$ prime that are inherited from conics of the Desarguesian plane $PG(2,q^t)$. Further, it is shown that certain conics do not inherit to arcs under the process of André replacement.

This short paper has two goals. First, explaining a simple procedure (which is essentially folklore) that, sometimes, makes it possible to obtain a formula for the number of solutions to a system of multivariate polynomial inequalities over a finite field. Second, applying that procedure to prove a formula for the number of contiguous superregular $4 \times 4$ matrices over a finite field. The formula was previously conjectured by Appuswamy, Bazzani, Connelly, Ekaireb, Congero, and Zeger [Probability of super-regular matrices and MDS codes over finite fields, arXiv:2603.20983]. In addition, the same procedure is used to provide formulas for the number of contiguous superregular $3 \times 4$, $3 \times 5$, and $3 \times 6$ matrices over a finite field.

This paper presents a random approach to the multibonacci sequence. We generalise the model introduced by Benjamin, Levin, Mahlburg, and Quinn, which is based on a random tiling method using dominoes and squares that leads to the Fibonacci sequence, and which was extended to the tribonacci case in a previous work by the authors. Our approach employs tiling with linear $k$-ominoes, $k=1,\ldots,s$, combined with specific colouring, to generate a weighted multibonacci sequence. For a natural random variable~$X$ defined by this model, we establish the distribution of $X$ in terms of multibonacci numbers and compute $\mathbb{E}[X] = 2^{s+1}-3$.

The asymmetric simple exclusion process (ASEP) is an important particle model with deep connections to orthogonal polynomials. Motivated by this connection, Corteel and Williams introduced the Koornwinder moments $M^{Z}_{\lambda}$ at $ t=q $, which generalize the moments of Askey–Wilson polynomials. They showed that the partition function of the two-species ASEP is equal to $M^{Z}_{\lambda}$ for a one-row partition $ \lambda $. In this paper, we investigate a conjecture of Rains on the positivity of the minimal numerator of the Koornwinder moment $M^{Z}_{\lambda}$. We derive the first explicit formula for this moment, thereby obtaining a precise formulation of the conjecture by determining the minimal denominator of $M^{Z}_{\lambda}$. We also propose a generalization of the conjecture for the more general Koornwinder moments $M^{Z}_{\lambda,\mu}$ indexed by two partitions at special parameter values. We prove the generalized Rains' conjecture in two special cases: $(\xi,q)=(1,0)$ and $(\xi,q)=(1,1)$. For $(\xi,q)=(1,0)$, we construct a lattice path model and obtain a combinatorial formula for $M^{Z}_{\lambda,\mu}$ in terms of non-intersecting lattice paths. For $(\xi,q)=(1,1)$, we establish an explicit product formula for $M^{Z}_{\lambda,\mu}$ and give a combinatorial interpretation using lecture hall tableaux.

Sum-rank codes provide a generalized framework for Hamming and rank-metric codes, with codewords represented as tuples of matrices and weight given by the sum of the block ranks. In this paper, we introduce and study a block-tensor-rank invariant for sum-rank metric codes. To each code, we associate its block tensor rank: the smallest number of block-simple tensors, namely rank-one matrices supported inside single blocks, whose linear span contains the code. In general, determining the block tensor rank of a sum-rank code is challenging. Our main structural result shows that the block tensor rank decomposes additively across the blocks of the code, thereby reducing its computation to a tensor-rank problem on each block projection. Consequently, we derive two complementary lower bounds on the block tensor rank, referred to as the projection-wise bound and the coordinate-code bound. Moreover, by combining the coordinate-code bound with the classical Singleton and Griesmer bounds for codes in the Hamming metric, we obtain explicit lower bounds, called the Singleton coordinate-code bound and the Griesmer coordinate-code bound, respectively. We further construct families of sum-rank codes whose block tensor ranks attain the Singleton or Griesmer coordinate-code bounds. These constructions are based on Hamming-metric codes achieving the corresponding classical bounds. Finally, we show that, in certain cases, the block tensor ranks of two known families of sum-rank codes in the literature do not attain the Singleton coordinate-code bound.

We determine all possible intersection arrays of completely regular codes of covering radius one in the generalized Petersen graphs \(GP(n,k)\), where \(n\geq 3\) and \(1\leq k<n/2\). In the equivalent language of perfect colorings, this amounts to enumerating all quotient matrices of perfect \(2\)-colorings, up to interchanging the two colors. Since \(GP(n,k)\) is cubic, there are only six possible nontrivial quotient matrices. For each of them, we give necessary and sufficient arithmetic conditions on \(n\) and \(k\) for its existence. The feasible cases are realized by explicit periodic colorings. The nonexistence part is obtained by reducing the local coloring conditions to cyclic systems of linear equations and applying a Fourier argument on roots of unity. Together with the previously known cases \(GP(n,2)\) and \(GP(n,3)\), the results give a complete arithmetic classification of quotient matrices, and hence of covering-radius-one completely regular code parameters, in the generalized Petersen family.

The classical Corrádi-Hajnal theorem states that for any multiple $n$ of $3$, if $G$ is a graph with $n$ vertices and $\delta(G) \geq 2n/3$, then $G$ can be partitioned into $n/3$ vertex-disjoint copies of the triangle [Acta Math. Acad. Sci. Hung., 14:423-439, 1964]. Balogh, Molla and Sharifzadeh obtained a smaller lower bound by adding the independence number condition [Random Struct. Algorithms, 49:669-693, 2016]. In this paper, we study perfect tilings in digraphs subject to conditions on the independence number and the degree. The independence number, $\alpha(D)$, of $D$ is the maximum integer $k$ such that $D$ has an independent set of cardinality $k$. We show that if $D$ is an $n$-vertex digraph with $\alpha(D)\leq o(1)n$ and $\delta(D) \geq (1+o(1))n$, then $D$ has a perfect $T_3$-tiling, where $T_3$ denotes a transitive triangle. This minimum degree condition is asymptotically best possible. Moreover, our result implies the theorem of Balogh, Molla, and Sharifzadeh concerning perfect triangle tilings.

Designing rapid transportation routes requires balancing efficiency and reachability. Shortest-path models ensure direct, cost-efficient routes but ignore coverage, while centrality-based approaches maximize accessibility but do not enforce operational constraints. We study the problem of selecting a shortest path that maximizes reachability, measured as the number of nodes within a fixed distance of the path. To do this, we introduce the $k$-Step-Central Shortest Path problem and analyse its structural properties. We show that optimal solutions on unweighted graphs can be found in polynomial time and propose an algorithm with a novel pruning rule. We also prove that the problem becomes NP-hard when edge weights are introduced. Additionally, we show that our algorithm can be used to solve the NP-hard problem of finding the closeness-central shortest path in a graph. We demonstrate the efficiency and scalability of our algorithm on synthetic and real-world networks with up to 2,000 nodes. Our results show that improving reachability can substitute for route expansion: increasing the reach of transit lines drastically increases their coverage with shorter routes. This suggests that investments in active transport infrastructure that improve reachability can be more effective than extending primary routes, providing a data-driven basis for allocating resources in network design.

Let $G = (V,E)$ be an edge-colored graph, and let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$. We show that for a sufficiently large $n$ if $G$ is an edge-colored triangle-free graph of order $n$ that satisfies $\delta^c(G)\geq (n+7)/5$, then $G$ contains a rainbow cycle of length four, which improves a bound of Ding et al. and is best possible. In addition, we show that given $k$, there is $n_0$ such that for $n\geq n_0$, if $G$ is an edge-colored triangle-free graph with $\delta^c(G)> n/5+3$, then $G$ contains a rainbow cycle of length $4k$.