--- title: Length-Constrained Expanders in Graph Theory url: https://www.emergentmind.com/topics/length-constrained-expanders type: topic --- # Length-Constrained Expanders in Graph Theory Searching arXiv for recent and foundational papers on length-constrained expanders and closely related formulations. {"query":"all:(\"length-constrained expander\" OR \"length-constrained expander decomposition\" OR \"long cycles in percolated expanders\" OR \"long induced paths in expanders\")","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} arxiv_search(query="all:(\"length-constrained expander\" OR \"length-constrained expander decomposition\" OR \"long cycles in percolated expanders\" OR \"long induced paths in expanders\")", max_results=10, sort_by="submittedDate", sort_order="descending") Length-constrained expanders are expansion objects in which a graph must satisfy not only a cut or congestion condition, but also a quantitative control on a length parameter. In current usage, the phrase refers to several related regimes: vertex-expansion at a prescribed scale forcing long paths or cycles, either deterministically or after percolation; algorithmic decompositions in which pairs initially within distance \(h\) must route over paths of length at most \(hs\); constant-hop expanders obtained by cut-matching games; explicit families indexed by every graph size with bounded rewiring cost; and high-dimensional or coding constructions in which the constrained quantity is the support length of cocycles or the local length of algebraic constraints [2407.11495] [1812.11562] [2404.13446] [1507.01196]. ## 1. Terminology and basic notions The classical vertex-expansion framework is explicit in the survey literature. A graph \(G=(V,E)\) on \(n\) vertices is an \(\alpha\)-expander if every \(U\subset V\) with \(|U|\le n/2\) satisfies \[ |N_G(U)|\ge \alpha |U|, \] where \(N_G(U)=\{v\in V\setminus U:\ v\text{ has a neighbor in }U\}\). The same survey isolates two local variants: \((I,\alpha)\)-expansion, which requires the inequality only for \(|U|\in I\), and \((k,\alpha)\)-expansion, which requires it for every \(|U|\le k\). It also proves that for every \(00\) such that a \((\{k\},\alpha)\)-expander on \(n\) vertices with \(k=\lfloor cn\rfloor\) contains an induced subgraph on at least \(c'n\) vertices that is an \(\alpha'\)-expander [1812.11562]. A more scale-specific formulation appears in the percolation setting. There, \(G\) is called an \((I,d)\)-expander if every \(X\subseteq V(G)\) with \(|X|\in I\) satisfies \[ |N(X)|\ge d|X|, \] and a \((\{k\},d)\)-expander is abbreviated as a \((k,d)\)-expander. The distinctive feature is that the condition may be imposed only at one exact size \(k\), not for all smaller sets [2407.11495]. This suggests that, for many length questions, the decisive parameter is not full multiscale expansion but the specific scale at which external-neighborhood growth is available. The algorithmic literature uses a different but related language. A length-constrained cut is a moving cut that increases edge lengths rather than deleting edges, and an \((h,s)\)-length \(\phi\)-expander decomposition is a cut \(C\) of size at most \[ |C|\le \kappa \phi |A| \] such that a node-weighting \(A\) is \((h,s)\)-length \(\phi\)-expanding in \(G-C\). Informally, nodes within distance \(h\) must be able to route flow over paths of length \(hs\) while using each edge only to an extent on the order of \(1/\phi\) [2404.13446]. The same survey literature places vertex expansion alongside edge and spectral formulations. For bounded-degree graphs, vertex expansion and edge expansion are qualitatively equivalent, and for a \(d\)-regular graph with second eigenvalue \(\lambda\), one has vertex-expansion parameter \[ \alpha=\frac{d-\lambda}{2d}. \] This link is important because several length results are proved under vertex expansion, while others are proved under strong spectral assumptions [1812.11562]. ## 2. One-scale vertex expansion and long structures after percolation The paper "Long cycles in percolated expanders" studies bond percolation on a deterministic host graph \(G\): the random subgraph \(G_p\) is obtained by retaining each edge independently with probability \(p\). Its central theorem for paths states that if every \(k\)-set satisfies \[ |N(S)|\ge kd \] and \[ p=\frac{1+\varepsilon}{d}, \] then, for sufficiently small constant \(\varepsilon>0\), the percolated graph \(G_p\) contains a path of length at least \[ \frac{\varepsilon^2kd}{10} \] with probability at least \[ 1-\exp\!\big(-\Omega_\varepsilon(kd)\big). \] The cycle theorem is the macroscopic analogue: if every set \(S\) of size \(cn/d\) satisfies \(|N(S)|\ge d|S|\), then \(G_p\) contains a cycle of order \(\Omega(\varepsilon^2 c^2 n)\) with probability at least \(1-\exp(-\Omega_{\varepsilon,c}(n))\). The proof gives the concrete lower bound \[ \frac{\varepsilon^2 c^2}{240}\,n \] after a two-round sprinkling argument [2407.11495]. The conceptual point is that one-scale vertex expansion already controls the supercritical behavior of \(G_p\). The path theorem uses DFS with deferred decisions. After \[ N_1=\frac{\varepsilon k d^2}{2(1+\varepsilon)} \] queries, the expected number of positive answers is \(\varepsilon kd/2\), and a Chernoff estimate implies that with exponentially high probability the search discovers enough vertices that either the active DFS stack already yields a path of length \(\varepsilon^2 kd/10\), or else the processed set forces too many edges between processed and unvisited vertices, contradicting the number of queries available. The cycle theorem then first extracts a path of length \[ \alpha n,\qquad \alpha=\frac{c\varepsilon^2}{40}, \] and uses a second exposure round to close a long segment of that path into a cycle [2407.11495]. This framework is weaker than ordinary global expansion. The hypothesis is only that every set of one exact size expands by factor \(d\) in external neighborhood. The paper explicitly contrasts this with edge expansion: edge-isoperimetric information alone cannot force long paths deterministically, as shown by highly unbalanced complete bipartite graphs such as \(K_{d,d^{10}}\), where small sets on the small side have large edge boundary but every path has length at most \(2d\). It also gives an application to \(K_{s,t}\)-free extremal graphs: if \(2\le s\le t\) and \(|E(G)|\ge \alpha n^{2-1/s}\), then there exists a sufficiently large constant \(K=K(\alpha,t)\) such that for any \[ p\ge \frac{K}{n^{1-1/s}}, \] the percolated graph \(G_p\) contains a cycle of length \(\Omega(n)\) with high probability [2407.11495]. ## 3. Deterministic length guarantees and cycle spectra In deterministic expander theory, weak local vertex expansion already forces long paths and cycles. The survey "Expanders - how to find them, and what to find in them" proves that if every \(k\)-set has at least \(\ell\) external neighbors, then the graph contains a path of length \(\ell\). It also proves that if every set \(W\) with \[ \frac{k}{2}\le |W|\le k \] satisfies \(|N_G(W)|\ge \ell\), then the graph contains a cycle of length at least \(\ell+1\). As corollaries, every \(\alpha\)-expander on \(n\) vertices contains a path of length at least \(\alpha n/2\) and, when \(n=\Omega(1/\alpha)\), a cycle of length more than \(\alpha n/4\) [1812.11562]. The same survey develops a much richer length spectrum. It shows that for every \(\alpha>0\) there exist constants \(A,C,n_0\) such that every \(\alpha\)-expander on \(n\ge n_0\) vertices and every target length \[ \ell\in [C\ln n,\ n/C] \] admits a cycle whose length lies between \(\ell\) and \(\ell+A\). Thus the cycle-length set is dense up to an additive constant across the full medium-length regime. The survey also sketches a weaker but still linear statement: an \(\alpha\)-expander on \(n\) vertices has \(\Theta(n)\) distinct cycle lengths. At the short end, ball-growth estimates imply diameter \(O_\alpha(\log n)\), and the paper remarks that expanders therefore have cycles “as short as \(O_\alpha(\log n)\)” [1812.11562]. A different deterministic length phenomenon concerns induced paths. The paper "Long induced paths in expanders" proves that if \(G\) is an \((n,d,\lambda)\)-graph with \[ \lambda < \frac{d^{3/4}}{100} \qquad\text{and}\qquad d<\frac n{10}, \] then \(G\) contains an induced path of length \[ \frac n{64\lambda}. \] More generally, the paper derives induced paths from an upper-uniformity condition on sparse graphs and packages the main engine as a two-graph theorem: if \(G'\subseteq G\) has minimum degree \(d\) and the host graph \(G\) satisfies two edge-distribution upper bounds on sets of sizes \(s_1,s_2,\ell\), then \(G'\) contains a path of length \(\ell\) that is induced in \(G\) [2402.02256]. These results delimit what weak expansion does and does not force. The survey explicitly does not claim pancyclicity, Hamiltonicity, or full intervals of consecutive cycle lengths for all expanders under weak hypotheses, and bipartite expanders already preclude odd-cycle guarantees [1812.11562]. Likewise, the induced-path theorem covers sufficiently strong spectral expanders, not arbitrary expanders in the loose \(\lambda\le (1-\varepsilon)d\) sense [2402.02256]. ## 4. Length-constrained expander decompositions and constant-hop expanders The algorithmic theory of length-constrained expansion replaces edge deletion by length increase. In this framework, a demand \(D\) is \(h\)-length if it is supported only on pairs at distance at most \(h\), a moving cut \(C\) increases edge lengths, and the \((h,s)\)-length sparsity of \(C\) with respect to a node-weighting \(A\) is defined by minimizing \[ \frac{|C|}{\operatorname{sep}_{hs}(C,D)} \] over all \(A\)-respecting \(h\)-length demands. The paper "New Structures and Algorithms for Length-Constrained Expander Decompositions" proves the routing characterization that if \(A\) is \((h,s)\)-length \(\phi\)-expanding, then every \(h\)-length \(A\)-respecting demand can be routed with congestion \(O(\log N/\phi)\) and dilation \(sh\), while failure of expansion yields a demand that cannot be routed with congestion \(1/(2\phi)\) and dilation \(sh/2\) [2404.13446]. Its main algorithmic theorem gives a witnessed \((\le h,s)\)-length \(\phi\)-expander decomposition in close-to-linear time. For \[ \epsilon \in \left[\frac1{\log^{1/c} N},\,1\right), \] it computes a decomposition with cut slack \[ \kappa=n^\epsilon \qquad\text{and}\qquad s=\exp(\operatorname{poly}(1/\epsilon)), \] using work \[ m\cdot \widetilde O\!\left(n^{\operatorname{poly}(\epsilon)}\operatorname{poly}(h)\right) \] and depth \[ \widetilde O\!\left(n^{\operatorname{poly}(\epsilon)}\operatorname{poly}(h)\right). \] The structural core is a union theorem: if \(C_1,\dots,C_k\) is a sequence of \((h,s)\)-length sparse moving cuts, then \(\sum_i C_i\) is again sparse, with \[ h' = 2h,\qquad s'=\frac{s-2}{2}, \] and a quantitative loss in \(\phi\) controlled by \[ 8^3\log^3 n\cdot n^{O(1/s)} \cdot \frac{\sum_i |C_i|}{\sum_i |C_i|/\phi_i}. \] A second pillar is a sparse-support flow algorithm with \[ |\operatorname{supp}(F)|\le \widetilde O(|E|+b), \] crucial for the recursive cutmatching machinery [2404.13446]. The later paper "Simple Length-Constrained Expander Decompositions" simplifies the existential side of this theory. It proves directly that every \(n\)-node, \(m\)-edge graph admits an \((h,s)\)-length \(\phi\)-expander decomposition of size \[ s\, n^{O(1/s)}\cdot \phi m, \] improving the earlier bound \[ \log n \cdot s\, n^{O(1/s)}\cdot \phi m. \] Its key quantitative improvement is a better loss for the union of sparse length-constrained cuts: \[ \log^3 n\cdot s^3 n^{O(1/s)} \quad\longrightarrow\quad s\cdot n^{O(1/s)}. \] This improvement is derived from a new arboricity bound \[ O(s\cdot n^{2/s}) \] for \(s\)-parallel-greedy graphs [2510.10227]. A parallel algorithmic line studies hop bounds directly. "A Cut-Matching Game for Constant-Hop Expanders" generalizes the cut-matching game so that the output is a \((t,2)\)-hop \(\phi\)-expander for unit-demands, with diameter at most \(t\), where \(t=1/\epsilon\), the number of cut rounds is \(O(1/\epsilon)\), and the final degree is \(n^{O(\epsilon)}\) up to polylogarithmic factors. Because the graph also has diameter at most \(t\), every unit-demand is \(t\)-hop, so the construction yields a genuine constant-hop expander when \(t=O(1)\): arbitrary unit-demands route over \(O(t)\)-hop paths with controlled congestion. The paper presents this as the bounded-length strengthening of ordinary expander embeddings, whose routings inherently use \(\Theta(\log n)\)-hop paths [2211.11726]. ## 5. Directed, vertex-capacitated, and shortcut generalizations The paper "Length-Constrained Directed Expander Decomposition and Length-Constrained Vertex-Capacitated Flow Shortcuts" extends the theory from undirected edge-capacitated graphs to two new settings. In the directed case, it defines \((h,s)\)-length \(\phi\)-expansion using \(h\)-length moving cuts and \(A\)-respecting symmetric \(h\)-length demands. It proves a routing characterization: if \(A\) is \((h,s)\)-length \(\phi\)-expanding, then every such demand can be routed with congestion \[ O\!\left(\frac{\log N}{\phi}\right) \] and length at most \(hs\); if not, some demand cannot be routed with congestion \(1/(2\phi)\) and length \(hs/2\). It also proves existence of a directed \((h,s)\)-length \(\phi\)-expander decomposition with cut slack \[ O\!\left(n^{O(1/s)}\log n\right) \] for \(s=O(\log n)\) [2503.23217]. For undirected vertex-capacitated graphs, the paper equips both vertices and edges with lengths and capacities, defines congestion as the maximum of edge and vertex congestion, and proves the corresponding routing theorem with the same \(O(\log N/\phi)\) congestion bound and dilation \(hs\). Existence of an \((h,s)\)-length \(\phi\)-expander decomposition again holds with cut slack \[ O\!\left(n^{O(1/s)}\log n\right). \] A structural reduction converts a vertex-capacitated graph \(G\) into a directed edge-capacitated graph \(G'\) by replacing each vertex \(v\) with \(v_{\mathrm{in}},v_{\mathrm{mid}},v_{\mathrm{out}}\). Distances are preserved, and expansion transfers in both directions up to a factor \(3\): if \(A\) is \((h,s)\)-length \(\phi\)-expanding in \(G\), then the corresponding \(A'\) is \((h,s)\)-length \(\phi\)-expanding in \(G'\), while the converse holds with parameter \(\phi/3\) [2503.23217]. The same paper builds length-constrained flow shortcuts for undirected vertex-capacitated graphs. For \[ \epsilon=\Omega(1/\log n), \] it constructs a \(t\)-step LC-flow shortcut \(E'\) of size \[ |E'|\le O\!\left(n^{1+O(\epsilon)}\log n/\epsilon^2\right), \] with length slack \[ O(1/\epsilon^3), \] congestion slack \[ O\!\left(n^{O(\epsilon)}\log^3 n/\epsilon^2\right), \] and step bound \[ t=2^{O(1/\epsilon)}. \] The guarantee has both directions. Forward mapping says that any flow feasible in \(G\) with congestion \(1\) and length \(h'\le h\) remains feasible in \(G\cup E'\) with congestion \(1\), length \(O(1/\epsilon^3)\cdot h'\), and at most \(2^{O(1/\epsilon)}\) edges per path. Backward mapping says that any flow in \(G\cup E'\) can be simulated in \(G\) with the same asymptotic length and the stated congestion blowup [2503.23217]. ## 6. Explicit size-constrained and algebraic constructions One concrete meaning of “length-constrained” concerns graph size rather than path length. "Explicit Expanding Expanders" constructs an infinite sequence of explicit \(d\)-regular multigraph expanders \[ G_0,G_1,G_2,\dots \] with \[ |V_{i+1}|=|V_i|+1. \] For any even \(d\ge 6\), the sequence starts at \(|V_0|=d/2+1\), has expansion cost at most \[ \frac{5d}{2}, \] and every graph in the sequence satisfies \[ h(G_i)\ge \frac d3 - O(\sqrt{d\log^3 d}). \] The paper also proves the general lower bound \[ \alpha(\mathcal G)\ge \frac{3d}{2} \] for any such family, and shows that intermediate graphs need not retain strong spectral expansion: for any \(\epsilon>0\), infinitely many members satisfy \[ \lambda_2(G)\ge \frac d2-\epsilon. \] Here the constrained parameter is the one-vertex growth step and the bounded rewiring needed to pass from size \(n\) to size \(n+1\) [1507.01196]. A different constrained regime is high-dimensional local spectral expansion. "High dimensional expanders and coset geometries" constructs finite pure \(n\)-dimensional simplicial complexes \(X^{(s)}\) as coset complexes of elementary matrix groups. For \(q>(n-1)^2\), every \(1\)-dimensional link has spectral gap at least \[ 1-\frac1{\sqrt q}, \] the entire complex is a one-sided local spectral expander with parameter \[ \frac1{\sqrt q-(n-1)}, \] and the \(1\)-skeleton has spectral gap at least \[ 1-\frac1{\sqrt q-(n-1)}. \] This is not a path-length theory, but it shows that strong expansion can coexist with rigid structural constraints such as bounded degree, \((n+1)\)-partiteness, clique-complex closure, and highly symmetric coset-incidence rules [1710.05304]. High-dimensional and coding papers shift the constrained quantity again. "Good Distance Lattices from High Dimensional Expanders" shows that bounded-degree high-dimensional cosystolic expanders over \(\mathbb Z\) yield lattices whose nonzero cohomology classes have large support: for every nontrivial cocycle \[ z\in Z^k(X;R)\setminus B^k(X;R), \qquad \|z\|\ge \mu. \] This turns expansion into a lower bound on the combinatorial length of lattice vectors [1803.02849]. In "New Codes on High Dimensional Expanders", coordinates are triangles of a \(2\)-dimensional expander, each edge-link has size \(q\), and local views are Reed–Solomon codewords on affine lines. The resulting code has dimension at least that of a Reed–Muller code with \(n/3\) variables and degree \(d\), has \(\Omega(|X_n(2)|)\) dimension when \(d>2q/3\), has constant relative distance when \(d