arXiv subjects

Gregory Gutin

Publications and source records attributed to Gregory Gutin.

At least 19 recordsLinked to original sources

arXiv2026

Hilton-Zhao vertex-splitting conjecture for triangle-free graphs

Every graph obtained from a regular Class $1$ graph by a vertex-splitting is overfull and hence is of Class $2$; the central question is whether the resulting graph is edge-chromatic critical. We consider a revised version of the Hilton-Zhao vertex-splitting conjecture for triangle-free base graphs, excluding the known cubic exception that arises from the Wagner graph. Let $G$ be a connected $n$-vertex triangle-free $Δ$-regular Class $1$ graph, and let $G^{*}$ be obtained from $G$ by splitting one vertex into two adjacent vertices. We first show that any non-extreme vertex-splitting produces a $Δ$-critical graph provided $Δ>\frac{n-1}{3}$. Consequently, for non-extreme vertex-splittings, the revised version of the conjecture is valid across its entire conjectured degree range, and indeed even under a slightly weaker degree condition. We next turn to arbitrary vertex-splittings, extreme ones included. It is shown that $G^*$ is $Δ$-critical provided $n \ge 10$ and $Δ>\frac{2n-3}{5}$. In addition, for $n>24$, the non-strict condition $Δ\ge \frac{2n-3}{5}$ is already sufficient. This extends the bipartite range $Δ>\frac{2n}{5}$ to the boundary range that the bipartite argument does not cover. In particular, our arguments give a complete treatment of the case of triangle-free $6$-regular base graphs in the conjectured range. The proofs bring together structural properties of full-deficiency pairs, homomorphism theorems for dense triangle-free graphs, and Hamilton decompositions built via graph detachment.

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 and introducing a one-sided completion lemma, we show that $h_\ell(G)\ge 2n/11$ for every $\ell$. Consequently, $f_{\mathrm{oe}}(G)\ge 2n/11$, 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 $f_o(G)\ge n/(2χ(G))$ can be replaced by $128/65<2$. Finally, a fourth-moment argument gives, for $α=α(G)\ge2$ and $δ(G)\ge 1$, \[ f_o(G)\ge \fracα{2}+\frac{\log_3α}{8} -\frac14\log_3\log_3\sqrtα. \] We also construct bipartite graphs satisfying \[ f_o(G)\le \frac{α(G)}2+\log_2\!\bigl(α(G)+1\bigr)+\frac12, \] showing that the logarithmic additive improvement over Scott's bound $f_o(G)\geα(G)/2$ has the optimal order of magnitude.

math.CO↗

arXiv2026

Small and large quasi-kernels: a weighted transfer theorem

A quasi-kernel of a digraph $D$ is an independent set from which every vertex is reachable by a directed path of length at most two. The Small Quasi-kernel Conjecture asserts that every source-free digraph on $n$ vertices has a quasi-kernel of size at most $n/2$, whereas the Large Quasi-kernel Conjecture asks for a quasi-kernel whose closed out-neighbourhood has size at least $n/2$. Spiro proved that the former implies the latter. In this paper, we establish a weighted transfer theorem: if every nonnegative integer weighting of a source-free digraph $D$ admits a quasi-kernel whose closed out-neighbourhood contains at least a $θ$-fraction of the total weight, then $D$ has a quasi-kernel of size at most $\lfloor(1-θ)α(D)+θμ\rfloor$ for $0<θ\le1/2$, where $μ$ is the maximum matching size between a maximum independent set and its complement. Using the independent blow-up construction of Spiro, we deduce that the universal small and large quasi-kernel assertions are equivalent for every such $θ$, with no loss in the parameter. In particular, the two conjectures are equivalent. As an application, we prove the half-order bound for source-free break digraphs.

math.CO↗

arXiv2026

Sharp bounds and extremal structure of minimally strongly $k$-connected digraphs

Let $D$ be a finite simple minimally strongly $k$-connected digraph, where $k\ge 2$, and let $n=|V(D)|$. Mader proved in 1985 that $|A(D)|\le 2k(n-k)$ for all sufficiently large $n$, with equality if and only if $D\cong \overleftrightarrow{K}_{k,n-k}$ (JCTB 1985). We prove that this bound holds whenever $n\ge 3k-1$ and that the order threshold $3k-1$ is best possible. Moreover, if $n\ge 3k$, then equality holds if and only if $D\cong \overleftrightarrow{K}_{k,n-k}$. The threshold $3k$ is also best possible. These results resolve the conjecture of Mader in (JCTB 1985).

math.CO↗

arXiv2026

Sharp lower bounds for bisections of graphs with bounded maximum degree

A bisection of a graph is a partition of its vertex set into two parts whose sizes differ by at most one. We prove that every graph with $m$ edges and maximum degree at most $k$ has a bisection of size at least $\frac{k+1}{2k}m$ when $k$ is odd, and at least $\frac{k+2}{2(k+1)}m$ when $k$ is even. Both bounds are sharp, with equality for complete graphs. This establishes the unweighted case of the bisection conjecture of Gerke, Gutin, Yeo and Zhou (arXiv 2024), removes the additive error from the corresponding bounds of Lee, Loh and Sudakov (JCTB 2013), and extends the odd-degree regular case proved by Bollobás and Scott (JGT 2004) to arbitrary graphs of bounded maximum degree. The proof combines the matching-and-pairing method with the Gallai--Edmonds decomposition. Hall's theorem controls the number of uncovered factor-critical components whose internal degrees are all even. A boundary-edge count bounds this obstruction by the total degree deficiency, while the remaining uncovered components supply the required pairing gain.

math.CO↗

arXiv2026

Note on near-spanning balanced antidirected trees missing from regular tournaments

We construct near-spanning balanced antidirected trees that fail to embed in highly symmetric tournaments despite the conjectured semidegree condition being satisfied. More precisely, for every sufficiently large odd integer $n$, we exhibit a regular vertex-transitive tournament on $n$ vertices and a balanced antidirected caterpillar on $n-1$ vertices with maximum degree at most $(1+o(1))n/\log_2 n$ that is not contained in the tournament. The host satisfies the strict $k/2$ threshold in both the semidegree and pseudo-semidegree settings, where $k$ is the number of arcs of the target tree. Consequently, this gives counterexamples to the uniform sublinear-degree interpretations of Conjectures~6.8 and~7.6 in Stein's survey (2024). The order $n-1$ of the target is best possible under the strict semidegree hypothesis. We also give a six-vertex balanced antidirected double-star missing from the seven-vertex Paley tournament, and show that seven is the smallest host order for a pseudo-semidegree counterexample with this fixed double-star.

math.CO↗

arXiv2026

Odd Properly Colored Cycles in Edge-Colored Graphs

It is well-known that an undirected graph has no odd cycle if and only if it is bipartite. A less obvious, but similar result holds for directed graphs: a strongly connected digraph has no odd cycle if and only if it is bipartite. Can this result be further generalized to more general graphs such as edge-colored graphs? In this paper, we study this problem and show how to decide if there exists an odd properly colored cycle in a given edge-colored graph. As a by-product, we show how to detect if there is a perfect matching in a graph with even (or odd) number of edges in a given edge set.

math.CO↗

arXiv2026

Exact Finite Attention Responses From RoPE Derivatives

We derive exact local responses for attention interventions, allowing candidate edits to be scored from a cached baseline and one backward pass. The starting point is the RoPE derivative $\partial_p z(p) = A z(p)$: its integral gives the finite positional displacement, which we carry through the softmax without linearising either rotation or normalisation. The resulting predictions achieve 95.36--96.52% sign accuracy across 92,160 executed positional edits on 768 held-out prompt sets, reducing answer-margin MAE by 73.6--82.5% against the positional Jacobian and by 36.2--50.9% against zero. For simultaneous key and value edits, the same divided-difference calculus isolates the interaction term $C_{KV} = \sum_j (p'_j - p_j)\,\varepsilon_j$, which is omitted by adding separate attributions. Retaining it reduces downstream margin MAE by more than a factor of nine in every setting of a 5,120-intervention sweep across two Qwen sizes, two tasks, and multiple layers; reductions against a quadratic interaction correction are 75.9--98.5%. Exactness concerns the edited attention write; downstream predictions contract that response with a baseline gradient and are evaluated by native execution. The calculus also yields a KL certificate for local approximation error, an exact query-conditioned gradient-step representation whose curvature identifies attention-preserving query directions, and minimum-norm query control. Sparse evaluation supports candidate ranking and cache decisions under explicit local distortion criteria.

stat.ML↗

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 Fernández, Kim, Kim and Liu proved that $f_2(n)=O(n)$, answering a question of Erdős, 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

Constructing edge-disjoint Steiner trees in Cartesian product networks

Cartesian product networks are always regarded as a tool for ``combining'' two given networks with established properties to obtain a new one that inherits properties from both. For a graph $F=(V,E)$ and a set $S\subseteq V(F)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a subgraph $T=(V',E')$ of $F$ that is a tree with $S\subseteq V'$. For $S\subseteq V(F)$ and $|S|\geq 2$, the {\it generalized local edge-connectivity} $λ(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $F$. For an integer $k$ with $2\leq k\leq n$, the {\it generalized $k$-edge-connectivity} $λ_k(F)$ of a graph $F$ is defined as $λ_k(F)=\min\{λ(S)\,|\,S\subseteq V(F) \ and \ |S|=k\}$.In this paper, we give sharp upper and lower bounds for $λ_k(G\Box H)$, where $\Box$ is the Cartesian product operation, and $G,H$ are two graphs.

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 $Δ(G)$ and chromatic index $χ'(G)$. A graph $G$ is called edge-chromatic $Δ$-critical if $χ'(G)=Δ(G)+1$ and $χ'(H)< χ'(G)$ for every proper subgraph $H$ of $G$, and $G$ is overfull if $\left|E(G)\right|>Δ(G)\lfloor |V(G)|/2\rfloor$. In 1986, Chetwynd and Hilton proposed the influential Overfull Conjecture: If $G$ is a simple graph with $Δ(G)>\frac{|V(G)|}{3}$, then $G$ is a Class $2$ graph if and only if $G$ contains an overfull subgraph $H$ with $Δ(H)=Δ(G)$. 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↗

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

On Seymour's and Sullivan's Second Neighbourhood Conjectures

For a vertex $x$ of a digraph, $d^+(x)$ ($d^-(x)$, resp.) is the number of vertices at distance 1 from (to, resp.) $x$ and $d^{++}(x)$ is the number of vertices at distance 2 from $x$. In 1995, Seymour conjectured that for any oriented graph $D$ there exists a vertex $x$ such that $d^+(x)\leq d^{++}(x)$. In 2006, Sullivan conjectured that there exists a vertex $x$ in $D$ such that $d^-(x)\leq d^{++}(x)$. We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and of triangle-free graphs. An oriented graph $D$ is an oriented split graph if the vertices of $D$ can be partitioned into vertex sets $X$ and $Y$ such that $X$ is an independent set and $Y$ induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when $Y$ induces a regular or an almost regular tournament.

math.CO↗

arXiv2026

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 $f_{oe}(G)$: $f_{oe}(G)\ge \lceil (n-1)/χ_{mm}{(G)} \rceil ,$ where $χ_{mm}{(G)}$ 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↗

arXiv2026

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 $δ(D) \ge n/ 2$, then $D$ contains a Hamilton oriented cycle with at least $δ(D)$ 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↗

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 $Δ(D)$ 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 $Δ(D)\leq~4$ 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 $Δ(D)\leq 3$ and $g(D)\in \{3,4,5\}$, then ${\rm fasd}(D)=g(D)$; (iii) If $Δ(D)\leq 3$ and $g(D)\ge 8$ then ${\rm fasd}(D)<g(D).$ We also give some bounds for the cases when $Δ$ or $g$ are large and state several open problems and a conjecture.

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 $Δ$. We prove the following bounds. If $D$ is an oriented graph, then $fvs(D)\leq \frac{3n}{7}$ when $Δ\le 4$ and $fvs(D)\leq \frac{n}{2}$ when $Δ\le 5$. If $D$ is a connected digraph, $Δ\le 4$ 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 $Δ\le 5$ then $fvs(D)\leq \frac{2n}{3}.$ Note that all the above bounds are tight.

math.CO↗