arXiv subjects

Gregory Gutin

Publications and source records attributed to Gregory Gutin.

At least 19 recordsLinked to original sources

arXiv2026

A linear bound for nested cycles without geometric crossings

Cycles $C_1,\ldots,C_k$ in a graph are called nested without geometric crossings if they are pairwise edge-disjoint, $V(C_k)\subseteq\cdots\subseteq V(C_1)$, and each pair of consecutive cycles induces the same cyclic order on the vertices of the inner cycle, up to reversal. Let $f_k(n)$ be the least number of edges that forces such a family in every $n$-vertex graph. Gil Fern\'andez, Kim, Kim and Liu proved that $f_2(n)=O(n)$, answering a question of Erd\H{o}s, and asked whether $f_k(n)=O_k(n)$ for every fixed $k$. We prove this for all $k$. The proof selects the inner cycles together with a disjoint subgraph that supplies their external neighbours. A reselection argument gives disjoint paths from every inner-cycle vertex to any sufficiently large target set. Sublinear expansion and a rooted clique minor then allow the vertices to be joined in the required cyclic order.

math.CO↗

arXiv2026

Bounds on Odd and Odd-Even Induced Subgraphs

Let $G$ be an $n$-vertex graph and let $\ell:V(G)\to\mathbb{F}_2$ prescribe degree parities. A set $S\subseteq V(G)$ is $\ell$-admissible if every $v\in S$ has degree congruent to $\ell(v)$ modulo $2$ in $G[S]$. Let $h_\ell(G)$ be the maximum order of an $\ell$-admissible set, set $f_{\mathrm{oe}}(G):=\min_\ell h_\ell(G)$, and write $f_o(G):=h_{\mathbf{1}}(G)$, where $\mathbf{1}(v)=1$ for every $v\in V(G).$ We prove three main results for graphs without isolated vertices. First, by extending Zeng's odd-cut method to arbitrary parity prescriptions an introducing a one-sided completion lemma, we show that $h_\ell(G)\ge n/6$ for every $\ell$. Consequently, $f_{\mathrm{oe}}(G)\ge n/6$, improving the previous bound $2n/21$. Second, for bipartite graphs we derive lower bounds on $f_o(G)$ in terms of the $\mathbb{F}_2$-rank of the bipartite adjacency matrix and combine them to obtain \[ f_o(G)\ge \left(\frac14+\frac1{256}\right)n=\frac{65}{256}n. \] Thus, in the bipartite case, the factor $2$ in Scott's bound $f_o(G)\ge n/(2\chi(G))$ can be replaced by $128/65<2$. Finally, writing $\alpha=\alpha(G)$, a fourth-moment argument gives, for $\alpha\ge2$, \[ f_o(G)\ge \frac{\alpha}{2}+\frac{\log_3\alpha}{8} -\frac14\log_3\log_3\sqrt{\alpha}. \] We also construct bipartite graphs satisfying \[ f_o(G)\le \frac{\alpha(G)}2+\log_2\!\bigl(\alpha(G)+1\bigr)+\frac12, \] showing that the logarithmic additive improvement over Scott's bound $f_o(G)\ge\alpha(G)/2$ has the optimal order of magnitude.

math.CO↗

arXiv2026

The Complexity of Mixed Arc-Disjoint Spanning Subdigraphs with Antistrong Connectivity

A trail is antidirected if its arcs alternate between forward and backward. A digraph $D$ is antistrong if, for every ordered pair of distinct vertices $x,y\in V(D)$, it contains a forward antidirected $(x,y)$-trail. Bang-Jensen, Bessy, Jackson and Kriesell [J. Combin. Theory Ser. B 122 (2017), 68--90] introduced antistrong connectivity and posed two problems concerning mixed arc-disjoint spanning subdigraphs. In the first problem, one seeks an antistrong spanning subdigraph and an arc-disjoint strong spanning subdigraph. In the second, strong connectivity is replaced by the requirement that the underlying graph of the second subdigraph be 2-edge-connected. Bang-Jensen et al. asked whether each of the two problems can be solved in polynomial time. We prove that the two associated decision problems are NP-complete. The first remains NP-complete for digraphs with maximum out-degree at most four and maximum in-degree at most five. The second remains NP-complete even for oriented digraphs that are strong and antistrong, whose underlying graphs are 3-vertex-connected, and in which all but at most two vertices have both in-degree and out-degree at most four. In particular, the latter hardness result does not rely on digons.

cs.DM↗

arXiv2026

Weighted Counting Formula and $2n/21$ Lower Bound for Induced Subgraphs with Prescribed Degree Parities

Let $G=(V,E)$ be a finite simple graph of order $n\geq 1$, and let $\ell:V\to\{0,1\}$ be a prescribed parity labeling. A set $S\subseteq V$ is called $\ell$-admissible if $d_S(v)\equiv \ell(v)\pmod 2$ for every $v\in S$, where $d_S(v)=|N_G(v)\cap S|$. Let $h_\ell(G)$ be the maximum order of an $\ell$-admissible set and let $f_{\rm oe}(G)=\min_\ell h_\ell(G)$. For $x\in\mathbb R$, define the weighted counting polynomial $$ M_{\ell,x}(G)=\sum_{S\in {\cal A}_\ell(G)}x^{|S|}, $$ where ${\cal A}_\ell(G)$ is the collection of all $\ell$-admissible sets in $G$. For $R\subseteq V$, let $z_\ell(R)$ be the number of vertices $v\in V\setminus R$ for which $d_R(v)\equiv\ell(v)\pmod 2$. We prove the exact identity $$ M_{\ell,x}(G) =2^{-n}\sum_{R\subseteq V} x^{|R|}(2+x)^{z_\ell(R)}(2-x)^{n-z_\ell(R)-|R|}. $$ If $G$ has no isolated vertices, then, for every $\ell$ and every $x\in(0,2)$, $ M_{\ell,x}(G)>x^{n/2}(4-x^2)^{n/4}. $ Combining this estimate with a binary-entropy upper bound and optimizing $x$ gives $$ f_{\rm oe}(G)>c_*n>\frac{2n}{21}, $$ where $c_*\approx0.095862615$. Ferber and Krivelevich (Adv. Math. 2022) proved that $h_{\mathbf{1}}(G)\ge 10^{-4}n$, where $\mathbf{1}$ is the all-one labeling. Since $h_{\mathbf{1}}(G)\ge f_{\rm oe}(G)$, our result improves coefficient in their bound by almost three orders of magnitude, and does so simultaneously for every labeling.

math.CO↗

arXiv2026

Edge-chromatic $4$-critical graphs and Overfull Conjecture for graphs with maximum degree $4$

Let $G$ be a simple graph with maximum degree $\Delta(G)$ and chromatic index $\chi'(G)$. A graph $G$ is called edge-chromatic $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ and $\chi'(H)< \chi'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>\Delta(G)\lfloor |V(G)|/2\rfloor$. In 1986, Chetwynd and Hilton proposed the influential Overfull Conjecture: If $G$ is a simple graph with $\Delta(G)>\frac{|V(G)|}{3}$, then $G$ is a Class $2$ graph if and only if $G$ contains an overfull subgraph $H$ with $\Delta(H)=\Delta(G)$. Motivated by the structural analysis for $4$-critical graphs (SIAM J. Discrete Math. 2019), we show more properties in this paper, especially four new forbidden configurations in any $4$-critical graph, and provide a new structural proof of Overfull Conjecture for graphs with maximum degree $4$.

math.CO↗

arXiv2025

Feedback vertex sets of digraphs with bounded maximum degree

A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$ makes it acyclic. Let $D$ be a digraph with $n$ vertices and maximum degree $\Delta$. We prove the following bounds. If $D$ is an oriented graph, then $fvs(D)\leq \frac{3n}{7}$ when $\Delta\le 4$ and $fvs(D)\leq \frac{n}{2}$ when $\Delta\le 5$. If $D$ is a connected digraph, $\Delta\le 4$ and $D$ is not obtained from an odd undirected cycle by replacing every edge with the pair of opposite arcs with the same endvertices, then $fvs(D)\leq \frac{n}{2}$. If $D$ is an arbitrary digraph with $\Delta\le 5$ then $fvs(D)\leq \frac{2n}{3}.$ Note that all the above bounds are tight.

math.CO↗

arXiv2025

Odd Induced Subgraphs in Graphs of Maximum Degree Four

A graph is called odd if all of its vertex degrees are odd. A long-standing conjecture asked whether there exists a positive constant $c$ such that every $n$-vertex graph without isolated vertices contains an odd induced subgraph on at least $cn$ vertices. In 2022, Ferber and Krivelevich resolved this conjecture affirmatively with $c=10^{-4}$. A natural question is to determine the largest possible constant $c$. In 1994, Caro remarked that if $2/7$ is a valid value for $c$, then it is the largest possible one. To the best of our knowledge, the bound $c\ge 2/7$ has not been improved. Previous research has established tight bounds for specific graph classes -- for instance, $c = 2/5$ for graphs with maximum degree at most $3$ and without isolated vertices. In this paper, we prove that $c=2/7$ is the tight bound for graphs with maximum degree at most $4$ and without isolated vertices. Our result provides some support for $2/7$ being the largest value of $c$.

math.CO↗

arXiv2025

Public Goods Games in Directed Networks with Constraints on Sharing

In a public goods game, every player chooses whether or not to buy a good that all neighboring players will have access to. We consider a setting in which the good is indivisible, neighboring players are out-neighbors in a directed graph, and there is a capacity constraint on their number, k, that can benefit from the good. This means that each player makes a two-pronged decision: decide whether or not to buy and, conditional on buying, choose which k out-neighbors to share access. We examine both pure and mixed Nash equilibria in the model from the perspective of existence, computation, and efficiency. We perform a comprehensive study for these three dimensions with respect to both sharing capacity (k) and the network structure (the underlying directed graph), and establish sharp complexity dichotomies for each.

cs.GT↗

arXiv2025

Note on Long Directed Cycles in Eulerian Digraphs

Huang, Ma, Shapira, Sudakov and Yuster (Comb. Prob. Comput. 2013) proved that every Eulerian digraph of average out-degree $d$ has a directed cycle of length at least $\sqrt{d}.$ We improve the lower bound from $\sqrt{d}$ to $\sqrt{2d}-3/2.$

math.CO↗

arXiv2025

Large induced subgraphs with prescribed degree parity

A long-standing conjecture of Caro (Discrete Math, 1994), confirmed by Ferber and Krivelevich (Adv Math, 2022), states that every $n$-vertex graph $G$ without isolated vertices contains an induced subgraph of order linear in $n$ in which every vertex has odd degree. We generalize this result to graphs $G$ whose vertices are labeled by $\ell: V(G)\to \{0,1\}$. We require, in an induced subgraph, all $0$-labeled vertices to have even degree and all $1$-labeled vertices to have odd degree. Let $h_{\ell}(G)$ denote the maximum order of such a subgraph. Let $f_{oe}(G)=\min_{\ell} h_{\ell}(G)$ be the worst-labeling parameter. We establish a pointwise lower bound for $h_{\ell}(G)$ that immediately yields a linear lower bound in $|V(G)|$ for $f_{oe}(G)$, where $G$ has no isolated vertices. For an $n$-vertex connected graph, we obtain a sharp lower bound for $f_{oe}(G)$: $f_{oe}(G)\ge \lceil (n-1)/{\chi}_{mm}{(G)} \rceil ,$ where ${\chi}_{mm}{(G)}$ is the maximum chromatic number of a minor of $G.$ Using proved cases of Hadwiger's Conjecture, we show that for $t\in \{3,4,5,6\}$, if an $n$-vertex connected graph $G$ is $K_t$-minor-free, then $f_{oe}(G)\ge \lceil (n-1)/(t-1)\rceil$ and this bound is sharp for each $t\in \{3,4,5,6\}$. Finally, we conjecture that $f_{oe}(G)\ge f_o(G)/2$ for all graphs $G$ and confirm the conjecture for all trees and complete multipartite graphs.

math.CO↗

arXiv2025

FPT Constant Approximation Algorithms for Colorful Sum of Radii

We study the colorful sum of radii problem, where the input is a point set $P$ partitioned into classes $P_1, P_2, \dots, P_\omega$, along with per-class outlier bounds $m_1, m_2, \dots, m_\omega$, summing to $m$. The goal is to select a subset $\mathcal{C} \subseteq P$ of $k$ centers and assign points to centers in $\mathcal{C}$, allowing up to $m_i$ unassigned points (outliers) from each class $P_i$, while minimizing the sum of cluster radii. The radius of a cluster is defined as the maximum distance from any point in the cluster to its center. The classical (non-colorful) version of the sum of radii problem is known to be NP-hard, even on weighted planar graphs. The colorful sum of radii is introduced by Chekuri et al. (2022), who provide an $O(\log \omega)$-approximation algorithm. In this paper, we present the first constant-factor approximation algorithms for the colorful sum of radii running in FPT (fixed-parameter tractable) time. Our contributions are twofold: We design an iterative covering algorithm that achieves a $(2+\varepsilon)$-approximation with running time exponential in both $k$ and $m$; We further develop a $(7+\varepsilon)$-approximation algorithm by leveraging a colorful $k$-center subroutine, improving the running time by removing the exponential dependency on $m$.

cs.CG↗

arXiv2025

Feedback Arc Sets and Feedback Arc Set Decompositions in Weighted and Unweighted Oriented Graphs

Let $D=(V(D),A(D))$ be a digraph with at least one directed cycle. A set $F$ of arcs is a feedback arc set (FAS) if $D-F$ has no directed cycle. The FAS decomposition number ${\rm fasd}(D)$ of $D$ is the maximum number of pairwise disjoint FASs whose union is $A(D)$. The directed girth $g(D)$ of $D$ is the minimum length of a directed cycle of $D$. Note that ${\rm fasd}(D)\le g(D).$ The FAS decomposition number appears in the well-known and far-from-solved conjecture of Woodall (1978) stating that for every planar digraph $D$ with at least one directed cycle, ${\rm fasd}(D)=g(D).$ The degree of a vertex of $D$ is the sum of its in-degree and out-degree. Let $D$ be an arc-weighted digraph and let ${\rm fas}_w(D)$ denote the minimum weight of its FAS. In this paper, we study bounds on ${\rm fasd}(D)$, ${\rm fas}_w(D)$ and ${\rm fas}(D)$ for arc-weighted oriented graphs $D$ (i.e., digraphs without opposite arcs) with upper-bounded maximum degree $\Delta(D)$ and lower-bounded $g(D)$. Note that these parameters are related: ${\rm fas}_w(D)\le w(D)/{\rm fasd}(D)$, where $w(D)$ is the total weight of $D$, and ${\rm fas}(D)\le |A(D)|/{\rm fasd}(D).$ In particular, we prove the following: (i) If $\Delta(D)\leq~4$ and $g(D)\geq 3$, then ${\rm fasd}(D) \geq 3$ and therefore ${\rm fas}_w(D)\leq \frac{w(D)}{3}$ which generalizes a known tight bound for an unweighted oriented graph with maximum degree at most 4; (ii) If $\Delta(D)\leq 3$ and $g(D)\in \{3,4,5\}$, then ${\rm fasd}(D)=g(D)$; (iii) If $\Delta(D)\leq 3$ and $g(D)\ge 8$ then ${\rm fasd}(D)<g(D).$ We also give some bounds for the cases when $\Delta$ or $g$ are large and state several open problems and a conjecture.

math.CO↗

arXiv2025

Oriented discrepancy of Hamilton cycles in oriented graphs satisfying Ore-type condition

Erd{\H o}s (1963) initiated extensive graph discrepancy research on 2-edge-colored graphs. Gishboliner, Krivelevich, and Michaeli (2023) launched similar research on oriented graphs. They conjectured the following extension of Dirac's theorem: If $D$ is an oriented graph on $n \ge 3$ vertices with minimum degree $\delta (D) \ge n/ 2$, then $D$ contains a Hamilton oriented cycle with at least $\delta(D)$ arcs in the same direction. This conjecture was proved by Freschi and Lo (2024) who posed an open problem to extend their result to an Ore-type condition. We propose two conjectures for such extensions and prove results which provide support to the conjectures.

math.CO↗

arXiv2024

Upper bounds on minimum size of feedback arc set of directed multigraphs with bounded degree

An oriented multigraph is a directed multigraph without directed 2-cycles. Let ${\rm fas}(D)$ denote the minimum size of a feedback arc set in an oriented multigraph $D$. The degree of a vertex is the sum of its out- and in-degrees. In several papers, upper bounds for ${\rm fas}(D)$ were obtained for oriented multigraphs $D$ with maximum degree upper-bounded by a constant. Hanauer (2017) conjectured that ${\rm fas}(D)\le 2.5n/3$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5. We prove a strengthening of the conjecture: ${\rm fas}(D)\le m/3$ holds for every oriented multigraph $D$ with $m$ arcs and maximum degree at most 5. This bound is tight and improves a bound of Berger and Shor (1990,1997). It would be interesting to determine $c$ such that ${\rm fas}(D)\le cn$ for every oriented multigraph $D$ with $n$ vertices and maximum degree at most 5 such that the bound is tight. We show that $\frac{5}{7}\le c \le \frac{24}{29} < \frac{2.5}{3}$.

math.CO↗

arXiv2024

Number of Subgraphs and Their Converses in Tournaments and New Digraph Polynomials

An oriented graph $D$ is converse invariant if, for any tournament $T$, the number of copies of $D$ in $T$ is equal to that of its converse $-D$. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph $D$ with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all converse invariant oriented graphs? In this paper, we introduce a digraph polynomial and employ it to give a necessary condition for an oriented graph to be converse invariant. This polynomial serves as a cornerstone in proving all the results presented in this paper. In particular, we characterize all orientations of trees with diameter at most 3 that are converse invariant. We also show that all orientations of regular graphs are not converse invariant if $D$ and $-D$ have different degree sequences. In addition, in contrast to the findings of El Sahili and Ghazo Hanna, we prove that every connected graph $G$ with maximum degree at least $3$, admits an orientation $D$ of $G$ such that $D$ is not converse invariant. We pose one conjecture.

math.CO↗

arXiv2024

On the $k$-anti-traceability Conjecture

An oriented graph is called $k$-anti-traceable if the subdigraph induced by every subset with $k$ vertices has a hamiltonian anti-directed path. In this paper, we consider an anti-traceability conjecture. In particular, we confirm this conjecture holds when $k\leq 4$. We also show that every sufficiently large $k$-anti-traceable oriented graph admits an anti-path that contains $n-o(n)$ vertices.

math.CO↗

arXiv2024

Bi-objective Optimization in Role Mining

Role mining is a technique used to derive a role-based authorization policy from an existing policy. Given a set of users $U$, a set of permissions $P$ and a user-permission authorization relation $\mahtit{UPA}\subseteq U\times P$, a role mining algorithm seeks to compute a set of roles $R$, a user-role authorization relation $\mathit{UA}\subseteq U\times R$ and a permission-role authorization relation $\mathit{PA}\subseteq R\times P$, such that the composition of $\mathit{UA}$ and $\mathit{PA}$ is close (in some appropriate sense) to $\mathit{UPA}$. In this paper, we first introduce the Generalized Noise Role Mining problem (GNRM) -- a generalization of the MinNoise Role Mining problem -- which we believe has considerable practical relevance. Extending work of Fomin et al., we show that GNRM is fixed parameter tractable, with parameter $r + k$, where $r$ is the number of roles in the solution and $k$ is the number of discrepancies between $\mathit{UPA}$ and the relation defined by the composition of $\mathit{UA}$ and $\mathit{PA}$. We further introduce a bi-objective optimization variant of GNRM, where we wish to minimize both $r$ and $k$ subject to upper bounds $r\le \bar{r}$ and $k\le \bar{k}$, where $\bar{r}$ and $\bar{k}$ are constants. We show that the Pareto front of this bi-objective optimization problem (BO-GNRM) can be computed in fixed-parameter tractable time with parameter $\bar{r}+\bar{k}$. We then report the results of our experimental work using the integer programming solver Gurobi to solve instances of BO-GNRM. Our key findings are that (a) we obtained strong support that Gurobi's performance is fixed-parameter tractable, (b) our results suggest that our techniques may be useful for role mining in practice, based on our experiments in the context of three well-known real-world authorization policies.

cs.CR↗

arXiv2024

Lower Bounds for Maximum Weight Bisections of Graphs with Bounded Degrees

A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. Our results improve a bound of Lee, Loh, and Sudakov (J. Comb. Th. Ser. B 103 (2013)) for (unweighted) maximum bisections in graphs whose maximum degree is either even or equals 3, and for almost all graphs. We show that a tight lower bound for maximum size of bisections in 3-regular graphs obtained by Bollobás and Scott (J. Graph Th. 46 (2004)) can be extended to weighted subcubic graphs. We also consider edge-weighted triangle-free subcubic graphs and show that a much better lower bound (than for edge-weighted subcubic graphs) holds for such graphs especially if we exclude $K_{1,3}$. We pose three conjectures.

math.CO↗