Largest $2$-regular Subgraphs in complete $S$-partite Graphs

Xudong Chen; Yiyang Jiang

arxiv: 2603.27424 · v3 · submitted 2026-03-28 · 🧮 math.CO

Largest 2-regular Subgraphs in complete S-partite Graphs

Yiyang Jiang , Xudong Chen This is my paper

Pith reviewed 2026-05-14 21:26 UTC · model grok-4.3

classification 🧮 math.CO

keywords 2-regular subgraphscomplete S-partite graphsinteger linear programmingefficient algorithmsgraph theorycontrollability

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The pith

The largest 2-regular subgraph in any complete S-partite graph is found by an integer linear program whose size depends only on S.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to locate the largest 2-regular subgraph inside a complete S-partite graph. Such a graph is built by expanding each vertex of S into an independent set and placing all possible edges between sets that are adjacent in S. The search is cast as an integer linear program whose variables and constraints are indexed only by the vertices of S. The program is solvable in cubic time in the number of vertices of S, so the running time stays fixed even when the partite sets grow arbitrarily large. This decouples the computational cost from the overall order of the graph.

Core claim

The integer linear program that selects a largest 2-regular subgraph of a complete S-partite graph admits an exact solution whose running time is O(|V(S)|^3) and therefore independent of the cardinalities of the partite sets.

What carries the argument

The integer linear program whose variables correspond to choices of edges between the partite sets dictated by S and whose constraints enforce 2-regularity.

If this is right

The maximum order of a 2-regular subgraph is computable from S and the part sizes alone.
The same optimum is achieved with high probability by a random S-partite graph with the same part sizes.
Large networks whose interconnection pattern is given by S can be analyzed for 2-regular substructures without scaling the algorithm to the full vertex count.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduction suggests that other subgraph problems whose feasibility depends only on the meta-graph S may also admit size-independent algorithms.
In structural systems theory the result supplies a concrete way to extract large controllable or stable subnetworks once the interconnection pattern S is known.
The same modeling step could be reused for directed or weighted versions of the problem provided the underlying partition type remains fixed.

Load-bearing premise

The integer linear program exactly encodes every feasible 2-regular subgraph of the complete S-partite graph.

What would settle it

For any small fixed S and concrete part sizes, enumerate all 2-regular subgraphs by brute force and compare their maximum size against the value returned by the integer program.

Figures

Figures reproduced from arXiv: 2603.27424 by Xudong Chen, Yiyang Jiang.

**Figure 1.** Figure 1: (a) Skeleton graph S. (b) The associated bipartite graph B. The highlighted edges show the correspondence between S and B. We also present the incidence matrices Z and Zˆ. Let Zˆ be the node-edge incidence matrix of B (without the normalization factor 1/2). We organize the rows of the matrix in a way that the ith (resp., (i+q)th) row, for 1 ≤ i ≤ q, corresponds to the node u ′ i (resp., u ′′ i ). Given the… view at source ↗

**Figure 2.** Figure 2: (a) The auxiliary pseudograph M for Example 1, with blue edges labeled by their order in E. (b) The corresponding complete S-partite graph Ky, with the lifted Hamilton cycle H shown in blue. Next, we make the following observation: Lemma 5. Let ui0 , . . . , uiL−1 be the sequence of nodes in the Euler circuit (12). Then, for any sequence of nodes vi0 , . . . , viL−1 such that vij ∈ π −1 (uij ) and vij ̸= v… view at source ↗

**Figure 3.** Figure 3: illustrates the above construction on a small instance. In this example, the skeleton in panel (a) has edge set {(u1, u1),(u1, u2),(u2, u3)} and capacity vector x = (3, 3, 3). An optimal coefficient vector satisfies c(u1, u1) = 3 and c(u2, u3) = 6, with all other coefficients equal to zero, so the support graph Sc shown in panel (b) has two nontrivial connected components. Panels (c) and (d) summarize the … view at source ↗

**Figure 4.** Figure 4: The skeleton graph S considered in the numerical study. For each chosen x, we sample a random S-partite graph G by retaining each admissible edge of Kx independently with probability p. Equivalently, G is drawn from a stochastic block model [6] with community sizes xi and edge-probability matrix given by p on the support of S and 0 elsewhere. For each sampled graph G, we define n(G) := max{|V (H)| | H ⊆ G … view at source ↗

**Figure 5.** Figure 5: Empirical PMFs Pb(n(G) = t) for the setting x := (24, 7, 4, 11, 6, 4) with n = 56. The dashed, vertical line is at n ∗ = 42. Values of ˆp ∗ for different p are reported. 198 200 202 204 206 208 0 0.2 0.4 0.6 0.8 1 value of p n ∗ = 210 t bP(n(G) = t) 4·log(n)/n n −0.4 6·log(n)/n 0.6 (a) Empirical PMFs. 207 208 209 210 211 0 0.2 0.4 0.6 0.8 1 100.0% 80.3% 59.5% 14.9% t (b) PMFs near n ∗ = 210 [PITH_FULL_IMA… view at source ↗

**Figure 6.** Figure 6: Empirical PMFs Pb(n(G) = t) for the case x = (120, 35, 22, 55, 30, 18) with n = 280. The dashed, vertical line marks n ∗ = 210, and the labels report ˆp ∗ for different p. small for p = n −0.4 , substantial for p = 4 log n/n, dominant for p = 6 log n/n, and nearly 1 for p = 0.6. We next consider a mid-size graph, with x = (120, 35, 22, 55, 30, 18) and n = 280, and let p ∈ n n −0.4 , 4 log n n , 6 log n n ,… view at source ↗

**Figure 7.** Figure 7: Plots of ˆp ∗ for different (k, p), with x = k(120, 35, 22, 55, 30, 18) and n = 280k, for k = 1, . . . , 5. 7 Conclusions In this paper, we have shown that the problem of finding largest 2-regular subgraphs in complete S-partite graphs can be solved efficiently, through the linear program (4). The extremal solutions of (4) are integer valued, and an integer solution can be obtained in time O(q 3 ), with q … view at source ↗

read the original abstract

In this paper, we focus on the class of complete $S$-partite graphs, for $S$ an undirected graph possibly with self-loops, and address the problem of finding largest $2$-regular subgraphs of these graphs, which can be formulated as an integer linear program. Roughly speaking, a complete $S$-partite graph is obtained by replacing every single node of $S$ with a number of nodes, preserving the edge/non-edge relations of $S$. Our motivation in finding largest $2$-regular subgraphs is rooted in the structural systems theory, particularly in the problem of finding largest subnetworks that can sustain controllability or asymptotic stability of the corresponding subsystems. A main contribution of the paper is to show that the integer linear problem can be solved efficiently in $O(|V(S)|^3)$, independent of the order/size of the $S$-partite graph itself. Furthermore, we demonstrate through simulations that with high probability, a random $S$-partite graph contains a largest $2$-regular subgraph of the same order as its complete counterpart does.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reduces largest 2-regular subgraph search on complete S-partite graphs to an ILP whose size depends only on |V(S)| and claims an O(|V(S)|^3) solver.

read the letter

The main thing to know is that the authors reduce the largest 2-regular subgraph problem on complete S-partite graphs to an integer linear program whose variables and constraints depend only on the base graph S, then solve it in cubic time in |V(S)|. This independence from the actual part sizes is the practical payoff when the parts are large. They also run simulations suggesting that random S-partite graphs often match the size achieved by the complete version. The motivation from structural controllability is stated clearly and gives the work a concrete application angle. The modeling step—tracking selected vertices per part and edges between parts—looks standard but is applied here to get the size reduction. The simulations are presented as supporting evidence, though without numbers or setup details they remain suggestive rather than conclusive. The soft spot is the linearity claim. The natural capacity constraints involve products of the per-part counts, which are bilinear, so the paper must supply an explicit linearization, a fixed-support enumeration, or a combinatorial reduction whose correctness is proved. If that step is missing or only sketched, the O(|V(S)|^3) bound does not follow directly from a standard ILP solver. The citation pattern is light and focused on the controllability literature, which is fine for this scope. This paper is for researchers who need polynomial algorithms for cycle-cover type problems on large multipartite graphs or who work on network controllability. A reader already familiar with ILP modeling of subgraphs will see the reduction technique as the useful piece. It deserves a serious referee because the runtime claim, if the linearization holds up, is a genuine algorithmic improvement for this graph family.

Referee Report

1 major / 1 minor

Summary. The paper studies largest 2-regular subgraphs in complete S-partite graphs (S an undirected graph possibly with loops), formulates the problem as an integer linear program whose variables aggregate vertices and edges by the parts of S, and claims that this ILP admits an exact solution in O(|V(S)|^3) time independent of the partite-set cardinalities n_i. It further reports simulation evidence that a random S-partite graph realizes the same maximum size with high probability.

Significance. A verified O(|V(S)|^3) algorithm would be useful in structural systems theory, permitting computation of maximal 2-regular subnetworks that preserve controllability or stability properties without scaling with total network order.

major comments (1)

[ILP formulation] ILP formulation section: the capacity constraints are written in the natural aggregated form x_ij ≤ k_i · k_j together with degree equations ∑_j x_ij = 2 k_i. These products are bilinear; the manuscript must supply an explicit linearization, a fixed-support enumeration over the k_i, or a combinatorial reduction (e.g., min-cost flow on an auxiliary digraph whose size depends only on |V(S)|) whose correctness is proved, otherwise the O(|V(S)|^3) claim cannot be substantiated.

minor comments (1)

[Simulations] The simulation section should report the precise distribution over S, the number of Monte-Carlo trials, and quantitative statistics (e.g., empirical probability and size gap) rather than the qualitative statement “with high probability.”

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the thorough review and for highlighting the issue with the ILP formulation. We agree that additional details are needed to justify the claimed time complexity.

read point-by-point responses

Referee: [ILP formulation] ILP formulation section: the capacity constraints are written in the natural aggregated form x_ij ≤ k_i · k_j together with degree equations ∑_j x_ij = 2 k_i. These products are bilinear; the manuscript must supply an explicit linearization, a fixed-support enumeration over the k_i, or a combinatorial reduction (e.g., min-cost flow on an auxiliary digraph whose size depends only on |V(S)|) whose correctness is proved, otherwise the O(|V(S)|^3) claim cannot be substantiated.

Authors: The referee correctly identifies that the bilinear constraints require further elaboration for the complexity claim to hold. In the revised manuscript, we will include an explicit reduction of the problem to a minimum-cost circulation problem on a directed graph with vertex set corresponding to the vertices of S (thus of size |V(S)|). Specifically, we construct an auxiliary network where nodes represent the parts, and arcs are designed to encode the possible edge usages x_ij and vertex counts k_i such that the flow conservation enforces the degree equations and the capacities are set to respect the part sizes n_i without introducing bilinear terms. The objective is encoded as the total flow or cost to maximize the covered vertices. This network has O(|V(S)|) vertices and O(|V(S)|^2) arcs, and can be solved in O(|V(S)|^3) time using standard min-cost flow algorithms (e.g., capacity scaling or cost scaling methods that are strongly polynomial). We will provide the full construction and proof of equivalence in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: ILP reduction and complexity bound are independent of target quantities

full rationale

The paper formulates largest 2-regular subgraphs in complete S-partite graphs as an integer linear program and claims this ILP can be solved in O(|V(S)|^3) time independent of partite set sizes. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation, or ansatz that is defined in terms of the output. The derivation chain relies on an external algorithmic reduction whose correctness is asserted separately from the input graph sizes, with no self-definitional or load-bearing self-citation patterns. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions of complete S-partite graphs and the equivalence between 2-regular subgraphs and cycle covers; no free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Standard definitions and properties of complete multipartite graphs and 2-regular subgraphs as disjoint unions of cycles.
Invoked when formulating the problem as an ILP on the structure of S.

pith-pipeline@v0.9.0 · 5483 in / 1218 out tokens · 31813 ms · 2026-05-14T21:26:16.461172+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the integer linear problem can be solved efficiently in O(|V(S)|^3), independent of the order/size of the S-partite graph itself
IndisputableMonolith/Foundation/ArithmeticFromLogic LogicNat ≃ Nat unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

reformulating the linear program (4) as an auxiliary optimization problem on a bipartite graph

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Convergence rate of $H$-property for step-graphons
math.PR 2026-04 unverdicted novelty 5.0

For step-graphons the probability of a node-disjoint cycle cover in the sampled graph converges to 0 or 1 at either an exponential rate or a root-n rate.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 1 Pith paper

[1]

Decentralized stabilization with symmetric topologies

A. Kirkoryan and M.-A. Belabbas. “Decentralized stabilization with symmetric topologies”. In:53rd IEEE Conference on Decision and Control. IEEE. 2014, pp. 1347–1352

work page 2014
[2]

Sparse linear ensemble systems and structural controllability

X. Chen. “Sparse linear ensemble systems and structural controllability”. In:IEEE Transac- tions on Automatic Control67.7 (2021), pp. 3337–3348

work page 2021
[3]

The traveling salesman problem: An overview of exact and approximate algo- rithms

G. Laporte. “The traveling salesman problem: An overview of exact and approximate algo- rithms”. In:European Journal of Operational Research59.2 (1992), pp. 231–247

work page 1992
[4]

Minimum-weight cycle covers and their approximability

B. Manthey. “Minimum-weight cycle covers and their approximability”. In:Discrete Applied Mathematics157.7 (2009), pp. 1470–1480

work page 2009
[5]

The Ring Star Prob- lem: Polyhedral Analysis and Exact Algorithm

M. Labb´ e, G. Laporte, I. Rodr´ ıguez-Mart´ ın, and J. J. Salazar-Gonz´ alez. “The Ring Star Prob- lem: Polyhedral Analysis and Exact Algorithm”. In:Networks43.3 (2004), pp. 177–189

work page 2004
[6]

Stochastic blockmodels: First steps

P. W. Holland, K. B. Laskey, and S. Leinhardt. “Stochastic blockmodels: First steps”. In: Social Networks5.2 (1983), pp. 109–137

work page 1983
[7]

Community Detection and Stochastic Block Models: Recent Developments

E. Abbe. “Community Detection and Stochastic Block Models: Recent Developments”. In: Journal of Machine Learning Research18.177 (2018), pp. 1–86

work page 2018
[8]

Limits of dense graph sequences

L. Lov´ asz and B. Szegedy. “Limits of dense graph sequences”. In:Journal of Combinatorial Theory, Series B96.6 (2006), pp. 933–957

work page 2006
[9]

Geometric Characterization of theH-Property for Step-Graphons

M.-A. Belabbas and X. Chen. “Geometric Characterization of theH-Property for Step-Graphons”. In:IEEE Transactions on Automatic Control69.6 (2024), pp. 3849–3864. 14

work page 2024
[10]

Blow-up Lemma

J. Koml´ os, G. N. S´ ark¨ ozy, and E. Szemer´ edi. “Blow-up Lemma”. In:Combinatorica17.1 (1997), pp. 109–123

work page 1997
[11]

Hamiltonicity of Step-graphons

X. Chen. “Hamiltonicity of Step-graphons”. In:arXiv:2510.02074(2025)

work page arXiv 2025
[12]

Edmonds, Matching and the Birth of Polyhedral Combinatorics

W. R. Pulleyblank. “Edmonds, Matching and the Birth of Polyhedral Combinatorics”. In: Documenta Mathematica(2012). Extra Volume: Optimization Stories, pp. 181–197

work page 2012
[13]

Largest 2-Regular Subgraphs in 3-Regular Graphs

I. Choi, R. Kim, A. V. Kostochka, B. Park, and D. B. West. “Largest 2-Regular Subgraphs in 3-Regular Graphs”. In:Graphs and Combinatorics35.4 (2019), pp. 805–813

work page 2019
[14]

Maximumr-Regular Induced Subgraph Problem: Fast Exponential Algorithms and Combinatorial Bounds

S. Gupta, V. Raman, and S. Saurabh. “Maximumr-Regular Induced Subgraph Problem: Fast Exponential Algorithms and Combinatorial Bounds”. In:SIAM Journal on Discrete Mathe- matics26.4 (2012), pp. 1758–1780

work page 2012
[15]

Hamiltonian cycles ink-partite graphs

L. DeBiasio, R. A. Krueger, D. Pritikin, and E. Thompson. “Hamiltonian cycles ink-partite graphs”. In:Journal of Graph Theory94.1 (2020), pp. 92–112

work page 2020
[16]

Finding Long Cycles in Balanced Tripartite Graphs: A First Step

G. Araujo-Pardo, Z. Berikkyzy, J. Faudree, K. Hogenson, R. Kirsch, L. Lesniak, and J. Mc- Donald. “Finding Long Cycles in Balanced Tripartite Graphs: A First Step”. In:Research Trends in Graph Theory and Applications. Ed. by D. Ferrero, L. Hogben, S. R. Kingan, and G. L. Matthews. Vol. 25. Association for Women in Mathematics Series. Cham: Springer, 2021,...

work page 2021
[17]

On theH-Property for Step-Graphons and Edge Polytopes

M.-A. Belabbas, X. Chen, and T. Ba¸ sar. “On theH-Property for Step-Graphons and Edge Polytopes”. In:IEEE Control Systems Letters6 (2022), pp. 1766–1771

work page 2022
[18]

On theH-property for Step-graphons: The Residual Case

W. Gao and X. Chen. “On theH-property for Step-graphons: The Residual Case”. In:IFAC- PapersOnLine59.4 (2025). 10th IFAC Conference on Networked Systems NECSYS 2025, pp. 7–12

work page 2025
[19]

Korte and J

B. Korte and J. Vygen.Combinatorial Optimization: Theory and Algorithms. 5th ed. Vol. 21. Algorithms and Combinatorics. Springer, 2012

work page 2012
[20]

Schrijver.Combinatorial Optimization: Polyhedra and Efficiency

A. Schrijver.Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003

work page 2003
[21]

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein.Introduction to Algorithms. 3rd ed. MIT Press, 2009

work page 2009
[22]

¨Uber die M¨ oglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren

C. Hierholzer and C. Wiener. “ ¨Uber die M¨ oglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren”. In:Mathematische Annalen6.1 (1873), pp. 30–32. 15

work page

[1] [1]

Decentralized stabilization with symmetric topologies

A. Kirkoryan and M.-A. Belabbas. “Decentralized stabilization with symmetric topologies”. In:53rd IEEE Conference on Decision and Control. IEEE. 2014, pp. 1347–1352

work page 2014

[2] [2]

Sparse linear ensemble systems and structural controllability

X. Chen. “Sparse linear ensemble systems and structural controllability”. In:IEEE Transac- tions on Automatic Control67.7 (2021), pp. 3337–3348

work page 2021

[3] [3]

The traveling salesman problem: An overview of exact and approximate algo- rithms

G. Laporte. “The traveling salesman problem: An overview of exact and approximate algo- rithms”. In:European Journal of Operational Research59.2 (1992), pp. 231–247

work page 1992

[4] [4]

Minimum-weight cycle covers and their approximability

B. Manthey. “Minimum-weight cycle covers and their approximability”. In:Discrete Applied Mathematics157.7 (2009), pp. 1470–1480

work page 2009

[5] [5]

The Ring Star Prob- lem: Polyhedral Analysis and Exact Algorithm

M. Labb´ e, G. Laporte, I. Rodr´ ıguez-Mart´ ın, and J. J. Salazar-Gonz´ alez. “The Ring Star Prob- lem: Polyhedral Analysis and Exact Algorithm”. In:Networks43.3 (2004), pp. 177–189

work page 2004

[6] [6]

Stochastic blockmodels: First steps

P. W. Holland, K. B. Laskey, and S. Leinhardt. “Stochastic blockmodels: First steps”. In: Social Networks5.2 (1983), pp. 109–137

work page 1983

[7] [7]

Community Detection and Stochastic Block Models: Recent Developments

E. Abbe. “Community Detection and Stochastic Block Models: Recent Developments”. In: Journal of Machine Learning Research18.177 (2018), pp. 1–86

work page 2018

[8] [8]

Limits of dense graph sequences

L. Lov´ asz and B. Szegedy. “Limits of dense graph sequences”. In:Journal of Combinatorial Theory, Series B96.6 (2006), pp. 933–957

work page 2006

[9] [9]

Geometric Characterization of theH-Property for Step-Graphons

M.-A. Belabbas and X. Chen. “Geometric Characterization of theH-Property for Step-Graphons”. In:IEEE Transactions on Automatic Control69.6 (2024), pp. 3849–3864. 14

work page 2024

[10] [10]

Blow-up Lemma

J. Koml´ os, G. N. S´ ark¨ ozy, and E. Szemer´ edi. “Blow-up Lemma”. In:Combinatorica17.1 (1997), pp. 109–123

work page 1997

[11] [11]

Hamiltonicity of Step-graphons

X. Chen. “Hamiltonicity of Step-graphons”. In:arXiv:2510.02074(2025)

work page arXiv 2025

[12] [12]

Edmonds, Matching and the Birth of Polyhedral Combinatorics

W. R. Pulleyblank. “Edmonds, Matching and the Birth of Polyhedral Combinatorics”. In: Documenta Mathematica(2012). Extra Volume: Optimization Stories, pp. 181–197

work page 2012

[13] [13]

Largest 2-Regular Subgraphs in 3-Regular Graphs

I. Choi, R. Kim, A. V. Kostochka, B. Park, and D. B. West. “Largest 2-Regular Subgraphs in 3-Regular Graphs”. In:Graphs and Combinatorics35.4 (2019), pp. 805–813

work page 2019

[14] [14]

Maximumr-Regular Induced Subgraph Problem: Fast Exponential Algorithms and Combinatorial Bounds

S. Gupta, V. Raman, and S. Saurabh. “Maximumr-Regular Induced Subgraph Problem: Fast Exponential Algorithms and Combinatorial Bounds”. In:SIAM Journal on Discrete Mathe- matics26.4 (2012), pp. 1758–1780

work page 2012

[15] [15]

Hamiltonian cycles ink-partite graphs

L. DeBiasio, R. A. Krueger, D. Pritikin, and E. Thompson. “Hamiltonian cycles ink-partite graphs”. In:Journal of Graph Theory94.1 (2020), pp. 92–112

work page 2020

[16] [16]

Finding Long Cycles in Balanced Tripartite Graphs: A First Step

G. Araujo-Pardo, Z. Berikkyzy, J. Faudree, K. Hogenson, R. Kirsch, L. Lesniak, and J. Mc- Donald. “Finding Long Cycles in Balanced Tripartite Graphs: A First Step”. In:Research Trends in Graph Theory and Applications. Ed. by D. Ferrero, L. Hogben, S. R. Kingan, and G. L. Matthews. Vol. 25. Association for Women in Mathematics Series. Cham: Springer, 2021,...

work page 2021

[17] [17]

On theH-Property for Step-Graphons and Edge Polytopes

M.-A. Belabbas, X. Chen, and T. Ba¸ sar. “On theH-Property for Step-Graphons and Edge Polytopes”. In:IEEE Control Systems Letters6 (2022), pp. 1766–1771

work page 2022

[18] [18]

On theH-property for Step-graphons: The Residual Case

W. Gao and X. Chen. “On theH-property for Step-graphons: The Residual Case”. In:IFAC- PapersOnLine59.4 (2025). 10th IFAC Conference on Networked Systems NECSYS 2025, pp. 7–12

work page 2025

[19] [19]

Korte and J

B. Korte and J. Vygen.Combinatorial Optimization: Theory and Algorithms. 5th ed. Vol. 21. Algorithms and Combinatorics. Springer, 2012

work page 2012

[20] [20]

Schrijver.Combinatorial Optimization: Polyhedra and Efficiency

A. Schrijver.Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003

work page 2003

[21] [21]

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein.Introduction to Algorithms. 3rd ed. MIT Press, 2009

work page 2009

[22] [22]

¨Uber die M¨ oglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren

C. Hierholzer and C. Wiener. “ ¨Uber die M¨ oglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren”. In:Mathematische Annalen6.1 (1873), pp. 30–32. 15

work page