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cs.DM

cs.DM: explore 112 source-linked works published from 2019 to 2026, with original documents and citations.

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Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

On the Extension Theorem for Packing Steiner Forests

We consider the problem of packing edge-disjoint Steiner forests in a graph. The input consists of a multi-graph $G=(V,E)$ and a collection of $t$ vertex subsets $S = \{S_1,S_2,\ldots,S_t\}$. A Steiner forest for $S$, also called an $S$-forest, is a forest of $G$ in which each $S_i$ is connected. In the case where $t=1$, this is the Steiner Tree packing problem. Kriesell's conjecture postulates that $2k$-edge-connectivity of $S_1$ is sufficient to find $k$ edge-disjoint $S_1$-trees. Lau showed that $24k$-edge-connectivity suffices for the Steiner Tree packing problem, which was improved to $6.5k$ by West and Wu and $5k+4$ by Devos, McDonald and Pivotto. In his thesis, Lau asserts that for the Steiner Forest problem, if each $S_i$ is $30k$-edge-connected in $G$, then there exist $k$ edge-disjoint $S$-forests. However, Lau's proof relies on an intermediate theorem called the Extension Theorem, which in this paper we will demonstrate has a gap by providing a counterexample to Lau's Extension Theorem. Furthermore, we will resolve this gap by correcting Lau's proof to show that $32k$-edge-connectivity of each $S_i$ suffices to pack $k$ $S$-forests. More careful analysis yields that $31k$-edge-connectivity of each $S_i$ is sufficient when $k \geq 8$.

cs.DM↗

arXiv2026

New Records for the Hadamard Maximal Determinant Problem

The Hadamard maximal determinant problem seeks an $n\times n$ matrix $X$ with entries in $\{\pm 1\}$, which maximizes the determinant $\vert \det X \vert$ for a given order $n \in \mathbb{N}$. We report matrices attaining new record determinants for orders $n\equiv 3\pmod{4}$ between $103$ and $119$, as well as $n=51$. We provide a proof of optimality within a specific family of circulant--block constructions for $X$. For these orders, we recast the Hadamard problem as a sequence reconstruction problem from a pair of periodic autocorrelation functions, subject to certain arithmetic conditions. We describe several learning--based strategies for reconstruction and provide a comparison with classical simulated annealing--style approaches as a benchmark.

math.CO↗

arXiv2026

Refutation of the Non-Cancelling Intersections Conjecture

The Non-Cancelling Intersections (NCI) conjecture of Amarilli, Monet and Suciu [arXiv:2401.16210] states that the union of a finite family of sets can always be built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In Wilhelm [arXiv:2608.19414] the conjecture was shown to fail when the witnessing dot-algebra expression is required to be left-linear. Here we remove that restriction and show that the conjecture is false in general: there is a finite lattice admitting no dot-algebra representation of its top element whatsoever. The counterexample is a lattice $P_{p,\mathfrak{m}}$ as in Wilhelm [arXiv:2608.19414], and the argument differs in only two ways. First, we replace the sequential "toggle game" of Wilhelm [arXiv:2608.19414] by a corresponding tree-shaped object, the plane tree, which stands to dot-algebra trees as the toggle game stands to left-linear ones. Second, we use a marked plane in which there is no admissible set of any size between $2p$ and $4p$, which also removes the need for the Erdős--Beck theorem and for the arithmetic Nullstellensatz. Consequently $p$ need not be astronomically large: every prime $p \ge 10^{5}$ works.

math.CO↗

arXiv2026

An Improved Bound for Smith's Longest Cycles Conjecture via a Forbidden Subdivision

Smith's conjecture asserts that in every $k$-connected graph with $k\geq 2$, any two longest cycles intersect in at least $k$ vertices. In this work, we establish an $Ω(k^{8/11})$ bound for this conjecture, improving upon the $Ω(k^{2/3})$ bound of Ma and Zhao. Our proof combines a Ramsey theoretic refinement of the traditional Turán-type approach with computer search.

math.CO↗

arXiv2026

Unary Functions, Automorphisms, and Unlabeled First-Order Model Counting

Every fixed first-order sentence $φ$ determines an enumerative sequence $n\mapsto\mathrm{FOMC}(φ,n)$, counting its models on the labeled domain $[n]$. We study the complexity of these sequences when logical specifications may use genuine unary function symbols and hence nested terms $x,f(x),f^2(x),\ldots$. We first prove that, for every fixed sentence $φ\in\mathrm{C}^1_{=}[f]$, with one unary function and an arbitrary finite relational vocabulary, $\mathrm{FOMC}(φ,n)$ is computable in time polynomial in $n$. By contrast, permitting either a second variable or a second unary function already yields hardness. Without counting quantifiers, there is a fixed sentence in $\mathrm{FO}^2_{=}[f]$ whose model-counting function is $\#\mathrm{P}_1$-complete. With one variable and two unary functions, there is a fixed constant-free universal sentence in $\mathrm{FO}^1_{=}[f,g]$, using only unary predicates besides $f$ and $g$, whose model-counting function is again $\#\mathrm{P}_1$-complete. We also relate labeled and unlabeled enumeration exactly. For every relational sentence $φ$, we construct an extension $φ_{\mathrm{aut}}$ in which a unary function records an automorphism and $\mathrm{FOMC}(φ_{\mathrm{aut}},n)=n!\cdot\mathrm{UFOMC}(φ,n)$, where $\mathrm{UFOMC}(φ,n)$ denotes the number of $n$-element models of $φ$ up to isomorphism. Thus automorphism marking gives a one-query exact reduction from unlabeled to labeled model counting at the same domain size. Over relational vocabularies of maximum arity at most $k$, where $k\geq2$, eliminating the auxiliary function yields single-query reductions from unlabeled $\mathrm{FO}^k_{=}$ and $\mathrm{C}^k$ model counting to labeled $\mathrm{FO}^{k+1}_{=}$ and $\mathrm{C}^{k+1}$ model counting, respectively.

cs.LO↗

arXiv2026

The Cayley Completion of a Graph

A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $γ^{+}$ (additions only) and the Cayley edit distance $γ_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $γ^{+}(G)\ge nΔ^{*}/(2m)-1$, where $Δ^{*}$ is the least $d\geΔ$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $γ^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $γ^{+}$, while $γ_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $γ^{+}(P_n)=γ^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $γ_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.

cs.DM↗

arXiv2026

Bounded Relative Boundary Implies Narrow DNF Approximation

Friedgut conjectured that an increasing family in the $p$-biased discrete cube with bounded relative boundary can be approximated arbitrarily well by one whose minimal elements have bounded size, with a bound independent of the dimension and the bias (J. Amer. Math. Soc. 12 (1999)). We prove this conjecture by showing that, for $0<p\leq 1/2$, every increasing Boolean function with total resampling influence at most $K$ is $\varepsilon$-close under $μ_p^n$ to a monotone DNF of width $\exp(O((K+1)^2/\varepsilon^2))$. A separate high-bias argument completes the proof for all $p\in(0,1)$. Our proof builds on Hatami's pseudo-junta theorem (Ann. of Math. 176 (2012)). Tracking Hatami's construction isolates an adaptive representation with increasing local activations and dimension-free arity and multiplicity-counted load bounds. Our main new ingredient is a bias-matched randomized shifting procedure that converts the pseudo-junta approximator into an increasing function while retaining exact measurability with respect to a controlled forced refinement of its adaptive representation. From the resulting monotone adaptive representation, we extract positive certificates and truncate them to obtain the required narrow DNF.

cs.CC↗

arXiv2026

Graphs with Long Pseudosimilarity Chains under Consecutive Vertex Deletions

Pseudosimilar vertices are vertices in distinct automorphism orbits whose deletions produce isomorphic graphs. Classical work has studied the existence, group-theoretic origin, and construction of large sets of such vertices. We ask a different recursive question: how long can one repeatedly delete a vertex that is pseudosimilar at the moment of deletion? We define the pseudosimilarity depth of a graph and construct connected graphs in which this process continues through all but a sublinear number of vertices. A two-clock construction gives a square-root deficit uniformly in the order, while a Chinese-remainder construction with many cyclic clocks yields an infinite family of asymmetric graphs with only a polylogarithmic number of vertices left outside the active chain. The mechanism realizes pseudosimilarity by breaking a long hidden automorphism orbit and enlarging the break one vertex at a time. Thus pseudosimilarity can persist through an asymptotically full sequence of vertex deletions, even though every graph encountered in the main construction is asymmetric.

math.CO↗

arXiv2026

On the Parameterized Complexity of $s$-Club Cluster Edge Deletion

We study the parameterized and kernelization complexity of the \emph{\textsc{$s$-Club Cluster Edge Deletion}} problem, a distance-bounded generalization of \emph{\textsc{Cluster Edge Deletion}}. Given a graph $G=(V,E)$ and integers $k,s$, the goal is to delete at most $k$ edges so that every resulting connected component has diameter at most $s$. On the structural side, we settle an open question of Montecchiani, Ortali, Piselli, and Tappini (\emph{Theoretical Computer Science}, 2023) by proving W[1]-hardness parameterized by pathwidth plus the maximum number of allowed $s$-clubs, and consequently by treewidth plus this parameter. Thus, the diameter bound $s$ is inecessary for tractability under these parameters. In contrast, we show that dependence on $s$ is unnecessary for several structural parameters: the problem is fixed-parameter tractable when parameterized by treedepth, neighborhood diversity, or cluster vertex deletion number, generalizing known results for $s=1.$ We further prove that no polynomial kernel exists when parameterized by vertex cover, even for $s=2$. On the positive side, we present an FPT bicriteria approximation scheme for graphs excluding long induced cycles, running in time $f(k,1/ε)\cdot n^{\mathcal{O}(1)}$ and producing a solution of size at most $k$ whose components have diameter at most $(1+ε)s$. Finally, we initiate the study of the directed variant, \textsc{$s$-Club Cluster Arc Deletion}, and prove that it is W[1]-hard parameterized by $k$, even on directed acyclic graphs.

cs.DM↗

arXiv2026

A Parametrized Complexity View on Robust Scheduling with Budgeted Uncertainty

In this study, we investigate a robust single-machine scheduling problem under processing time uncertainty. The uncertainty is modeled using the budgeted approach, where each job has a nominal and deviation processing time, and the number of deviations is bounded by Γ. The objective is to minimize the number of tardy jobs where a job is considered tardy if there is some scenario in which it is completed after its due date. Since the problem is NP-hard in general, we focus on analyzing its tractability under the assumption that certain natural parameters of the problem are each bounded by a constant. We consider three parameters: the robustness parameter Γ, the number of distinct due dates in the instance, and the number of jobs with nonzero deviations. Using parameterized-complexity theory, we prove that the problem is W[1]-hard with respect to Γ, but can be solved in XP time with respect to the same parameter. With respect to the number of distinct due dates, we establish a stronger hardness result by showing that the problem remains NP-hard even when there are only two different due dates and is solvable in pseudo-polynomial time when the number of due dates is upper bounded by a constant. To complement these results, we show that the case of a common (single) due date reduces to a robust binary knapsack problem with equal item profits, a problem we prove to be solvable in polynomial time. Finally, we prove that the problem is fixed-parameter tractable with respect to the number of jobs with nonzero deviations.

cs.DM↗

arXiv2026

From b-Coloring to $b^*$-Coloring: Large Girth and Parameterized Complexity

A b-coloring is a proper vertex coloring such that every color class contains a vertex, a so-called b-vertex, which sees all colors in its closed neighborhood. This type of coloring has been intensively studied from both structural and algorithmic point of view. Recently, Zaker [DAM 2025] introduced the notion of a b*-coloring, which is a b-coloring in which there is a vertex that sees a b-vertex of every color in its closed neighborhood. The b*-chromatic number is the maximum integer k such that there is a b*-coloring with k colors. We partially answer a question posed by Zaker and prove that graphs of girth at least 7 are b*-monotonic, which means that the b*-chromatic number does not increase by taking an induced subgraph. In addition, we discover a class of d-regular graphs of girth at least 5 with b*-chromatic number d+1, which strengthens a result about b-colorings by Dettlaff, Furmańczyk, Peterin, Roux, and Ziemann [AMC 2024]. We also study the parameterized complexity of finding b*-colorings, and show that for many structural parameters, the complexity coincides with that of finding b-colorings. In particular, the b*-chromatic number can be computed in polynomial time on any class of bounded clique-width. For most parameters, the translation from b-colorings is straightforward but for the feedback edge number, the FPT algorithm for b*-colorings is actually much simpler than that for b-colorings by Balabán [MFCS 2026].

cs.DM↗

arXiv2026

Ordered Ruzsa-Szemeredi Numbers at Matching Size Two

Bondy and Szwarcfiter defined $\mathrm{ex}^*(n,F)$ as the largest number of edges in an $n$-vertex graph whose edge set partitions into induced copies of $F$; for $F=2K_2$ the deficiency $\binom{n}{2}-\mathrm{ex}^*(n,2K_2)$ is $Θ(n^{3/2})$. We study the ordered relaxation at fixed matching size, in which each part need only be induced in the union of itself with the parts that follow it; write $\mathrm{ORS}_n(r)$ for the largest number of parts, so that $r\,\mathrm{ORS}_n(r)$ is the ordered analogue of $\mathrm{ex}^*(n,rK_2)$. Our main tool is a characterisation valid for every $r$: an ordered decomposition into induced $r$-matchings is a sequence of steps that start from $K_n$ and repeatedly delete a perfect matching from $2r$ vertices currently spanning a clique. Reading a decomposition backwards turns a condition about the ordering into a reachability question that an exhaustive search can settle. For $r=2$ we determine $\mathrm{ORS}_n(2)$ exactly at orders five through nineteen, where it takes the values $1,3,5,8,11,14,19,23,28,34,40,47,54,62,70$, and we confine $\mathrm{ORS}_{20}(2)$ to $\{78,79\}$. The counting bound $\lfloor n(n-4)/4\rfloor$ is attained at orders five through nine and at eleven, and missed by exactly one part at every other order below twenty, so order eleven is an isolated exception, not a parity effect. Across this range the ordered deficiency equals $\frac32n+O(1)$, and along powers of two a dyadic construction keeps it below $O(n\log n)$; whether it is linear for all $n$ is our main open question. The structural results are formalised in Lean 4, and the searches are certified by fail-closed sweeps and an independent checker.

math.CO↗

arXiv2026

A General Plotkin-type Bound on Function-Correcting Codes with Wyner-Graham Distance

Function correcting codes (FCCs) are designed to protect a specified function evaluation of messages at a higher level than the level of protection for messages, against errors while reducing the redundancy required for reliable communication. FCCs have thus far been studied for channels matched to various distances, including the Hamming and Lee distances. Every function partitions the message space into preimage sets corresponding to its distinct function values. Existing Plotkin-type bounds on the optimal redundancy of FCCs under the studied distances, applicable to arbitrary functions on the message space, depend on the pairwise distances among all the message vectors. This makes these bounds difficult to compute. We derive a general Plotkin-type bound on the optimal redundancy of FCCs under Wyner Graham distances, which include the Hamming and Lee distances as special cases. Our bound depends only on the cardinalities of the preimage sets and the sum of pairwise distances only among vectors within each preimage set. This approach significantly reduces the computations required to evaluate the existing Plotkin-type bounds and yields simplified bounds that are easier to compute for specific functions. We obtain simplified Plotkin-type bound for linear functions under the Wyner-Graham distance. Furthermore, the existing simplified bounds for linear functions under the Hamming and Lee distances are recovered as special cases of the proposed bound. We also obtain simplified bounds for several important classes of functions, including the Hamming weight function, the Hamming weight distribution function, the monomial functions under the Hamming distance, and the modular sum function, the Lee weight function, and the Lee weight distribution function under the Lee distance.

cs.IT↗

arXiv2026

FirstFit online coloring in the random order model

The average performance of FirstFit online coloring on trees in the random order model is completely determined in recent works of Frei et al. and Bosek et al., showing $Θ(\log n /\log\log n)$ number of colors, improving the $Θ(\log n)$ colors in the adversarial model. We provide a few further results on slightly more general graph classes. Firstly, we extend their method to obtain a simple path-counting principle for sparse graph classes, which immediately yields for example that cactus graphs and uniform hypertrees exhibit a similar improvement. We then show that FirstFit uses only $O(1)$ colors on crown graphs, a standard example where adversarial arrival forces $Θ(n)$ colors. We further show that density alone (even linear minimum degree) is insufficient to guarantee $O(1)$ colors even on bipartite graphs. Finally, we identify graph classes, including unit interval graphs and some graphs of high chromatic number, for which random arrival provides only limited improvement. We end with some open problems.

cs.DS↗

arXiv2026

Fair Division of Graphs: Beyond Traceability

In this paper, we study fair division problems in which resources are structured as graphs and agents must receive connected bundles. This connectivity requirement fundamentally alters the problem, making it significantly more challenging than its classical counterpart. We focus on the fairness notion of $\mathrm{EF1}_{\mathrm{outer}}$, where envy can be eliminated by removing at most one vertex whose deletion does not disconnect the bundle -- a critical constraint for applications such as land division and network allocation. Our first result extends prior work by establishing the existence of $\mathrm{EF1}_{\mathrm{outer}}$ allocations for an infinite family of non-traceable graphs (that is, graphs that do not admit a Hamiltonian path), answering a central open question and generalizing Bilò et al.'s result for traceable graphs. We then make progress on a conjecture concerning the $\mathrm{EF1}_{\mathrm{outer}}$ spectrum of trees due to Chen and Zwicker. Finally, we complement our structural results with algorithmic insights, showing that deciding the existence of an $\mathrm{EF1}_{\mathrm{outer}}$ allocation is NP-complete even for binary additive valuations, thereby resolving an open complexity question. Taken together, our results deepen the connection between graph theory and fair division, and offer new tools for studying fairness in structured resource environments.

cs.GT↗

arXiv2026

Constructive Characterization and Recognition Algorithm for Grafts with a Connected Minimum Join

Minimum joins in a graft $(G, T)$, also known as minimum $T$-joins of a graph $G$, are said to be connected if they determine a connected subgraph of $G$. Grafts with a connected minimum join have gained interest ever since Middendorf and Pfeiffer showed that they satisfy Seymour's min-max formula for joins and $T$-cut packings; that is, in such grafts, the size of a minimum join is equal to the size of a maximum packing of $T$-cuts. In this paper, we provide a constructive characterization of grafts with a connected minimum join. We also obtain a polynomial time algorithm that decides whether a given graft has a connected minimum join and, if so, outputs one. Our algorithm has two bottlenecks; one is the time required to compute a minimum join of a graft, and the other is the time required to solve the single-source all-sink shortest path problem in a graph with conservative $\pm 1$-valued edge weights. Thus, our algorithm runs in $O(n(m + n\log n) )$ time. In the nondense case, it improves upon the time bound for this problem due to Sebő and Tannier that was introduced as an application of their results on metrics on graphs.

cs.DM↗

arXiv2026

Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices

In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \times m$ minors equals $Δ$.

math.CO↗

arXiv2026

A Simplified Analysis of the Good-Bad $3/2$-Approximation Algorithm for Some Minimum-Cost Graph Problems

In this paper, we consider an easy greedy approximation algorithm, the good-bad algorithm, introduced by Couëtoux for finding a minimum-cost set of edges such that every connected component has at least $k$ vertices. Couëtoux proves that the good-bad algorithm achieves a $3/2$-approximation for this problem. Davis and Williamson extend this result to the more general problem of finding a minimum-cost edge set that contains at least one edge from every cut $S\subseteq V$ satisfying $h(S) = 1$ where $h:2^V \rightarrow \{0,1\}$ is downward monotone; that is, $h(S) = 1$ implies $h(T) = 1$ for every nonempty subset $T \subseteq S$. The original problem corresponds to $h(S) =1$ when $|S|<k$. We give a simplified analysis of the good-bad algorithm for downward monotone functions.

cs.DS↗

Compare source metadata on this page

Work	Published	Source identifier	Source
On the Extension Theorem for Packing Steiner Forests	2026-08-31	2603.16956	arxiv
New Records for the Hadamard Maximal Determinant Problem	2026-10-01	2608.22518	arxiv
Refutation of the Non-Cancelling Intersections Conjecture	2026-08-31	2608.27416	arxiv
An Improved Bound for Smith's Longest Cycles Conjecture via a Forbidden Subdivision	2026-08-31	2608.30353	arxiv
Unary Functions, Automorphisms, and Unlabeled First-Order Model Counting	2026-08-31	2608.30580	arxiv
The Cayley Completion of a Graph	2026-08-31	2608.30894	arxiv
Bounded Relative Boundary Implies Narrow DNF Approximation	2026-08-31	2609.00240	arxiv
Graphs with Long Pseudosimilarity Chains under Consecutive Vertex Deletions	2026-08-31	2609.00394	arxiv
On the Parameterized Complexity of $s$-Club Cluster Edge Deletion	2026-08-30	2510.07065	arxiv
A Parametrized Complexity View on Robust Scheduling with Budgeted Uncertainty	2026-08-30	2602.02748	arxiv
From b-Coloring to $b^*$-Coloring: Large Girth and Parameterized Complexity	2026-08-30	2608.03819	arxiv
Ordered Ruzsa-Szemeredi Numbers at Matching Size Two	2026-08-30	2608.14695	arxiv
A General Plotkin-type Bound on Function-Correcting Codes with Wyner-Graham Distance	2026-08-30	2608.29573	arxiv
FirstFit online coloring in the random order model	2026-08-30	2608.29603	arxiv
Fair Division of Graphs: Beyond Traceability	2026-08-30	2608.29902	arxiv
Constructive Characterization and Recognition Algorithm for Grafts with a Connected Minimum Join	2026-08-29	2510.26975	arxiv
Column Number of Delta-modular matrices: Refined Analysis via Sauer Matrices	2026-08-29	2608.06215	arxiv
A Simplified Analysis of the Good-Bad $3/2$-Approximation Algorithm for Some Minimum-Cost Graph Problems	2026-08-29	2608.29274	arxiv

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