arXiv subjects

Arash Beikmohammadi

Publications and source records attributed to Arash Beikmohammadi.

5 recordsLinked to original sources

arXiv2026

Bounds for the Vertex Chromatic Number of Connected Triangle-Free Graphs

It was recently shown that every connected graph of order $n \geq 5$ and size $m$ satisfies $χ(G) \leq \left\lceil \frac{m}{\sqrt{n}} \right\rceil$, and it was asked whether the stronger inequality $χ(G) \leq \left\lceil \frac{m}{2\sqrt{n}} \right\rceil + 1$ holds for every connected triangle-free graph. In this paper, we answer this question in the affirmative. In fact, we prove that every connected triangle-free graph $G$ with $G \not\cong C_5$ satisfies $χ(G) \leq \left\lceil \frac{m}{\sqrt{5.5n}} \right\rceil + 1$, where the constant $\sqrt{5.5}$ cannot be replaced by any constant greater than or equal to $\sqrt{6}$, and the equality holds for the Grötzsch graph and for every odd cycle of length between $7$ and $21$.

math.CO↗

arXiv2025

Discrete Homotopy and Promise Constraint Satisfaction Problem

The Promise Constraint Satisfaction Problem (PCSP for short) is a generalization of the well-studied Constraint Satisfaction Problem (CSP). The PCSP has its roots in such classic problems as the Approximate Graph Coloring and the $(1+\varepsilon)$-Satisfiability problems. The area received much attention recently with multiple approaches developed to design efficient algorithms for restricted versions of the PCSP, and to prove its hardness. One such approach uses methods from Algebraic Topology to relate the complexity of the PCSP to the structure of the fundamental group of certain topological spaces. In this paper, we attempt to develop a discrete analog of this approach by replacing topological structures with combinatorial constructions and some basic group-theoretic concepts. We consider the `one-dimensional' case of the approach. We introduce and prove the basics of the framework, show how it is related to the complexity of the PCSP, and obtain several hardness results, including the existing ones, as applications of our approach. The main hope, however, is that this discrete variant of the topological approach can be generalized to the `multi-dimensional' case needed for further progress.

cs.CC↗

arXiv2025

Tight Bounds for Cycle-Edge Decompositions and Covers

An old conjecture of Erd{ő}s and Gallai states that every $n$ vertex graph can be decomposed, that is $E(G)$ can be partitioned, into $O(n)$ cycles and edges. The covering version of this conjecture was proven by Pyber in 1985, where it was shown that all graphs can be covered by $n-1$ cycles and edges. The best upper bound on the number of cycles and edges required to decompose any graph is $O(n\log^*(n))$, which was recently shown by Buci{ć} and Montgomery in 2023. Here $\log^*(n)$ denotes the iterated logarithm function. Meanwhile, a construction of Erdős demonstrate that there exists graphs which require $(\frac{3}{2}-o(1))n$ cycles and edges to be decomposed. We prove all graphs with maximum degree at most $4$ can be decomposed into $n-1$ or fewer cycles and edges. We also show that every $n$ vertex claw-free graph can be decomposed into $n-1$ or fewer $2$-regular subgraphs and edges. Finally, we prove that every graph $G$ containing a cycle can be covered by $n-2$ or fewer cycles and edges. This improves Pyber's covering theorem by proving that $n-1$ cycles and edges are required only for trees.

math.CO↗

arXiv2021

On the Chromatic Vertex Stability Number of Graphs

The chromatic vertex (resp.\ edge) stability number ${\rm vs}_χ(G)$ (resp.\ ${\rm es}_χ(G)$) of a graph $G$ is the minimum number of vertices (resp.\ edges) whose deletion results in a graph $H$ with $χ(H)=χ(G)-1$. In the main result it is proved that if $G$ is a graph with $χ(G) \in \{ Δ(G), Δ(G)+1 \}$, then ${\rm vs}_χ(G) = {\rm ivs}_χ(G)$, where ${\rm ivs}_χ(G)$ is the independent chromatic vertex stability number. The result need not hold for graphs $G$ with $χ(G) \le \frac{Δ(G)+1}{2}$. It is proved that if $χ(G) > \frac{Δ(G)}{2}+1$, then ${\rm vs}_χ(G) = {\rm es}_χ(G)$. A Nordhaus-Gaddum-type result on the chromatic vertex stability number is also given.

math.CO↗

arXiv2021

On the chromatic edge stability index of graphs

Given a non-trivial graph $G$, the minimum cardinality of a set of edges $F$ in $G$ such that $χ'(G \setminus F)<χ'(G)$ is called the chromatic edge stability index of $G$, denoted by $es_{χ'}(G)$, and such a (smallest) set $F$ is called a (minimum) mitigating set. While $1\le es_{χ'}(G)\le \lfloor n/2\rfloor$ holds for any graph $G$, we investigate the graphs with extremal and near-extremal values of $es_{χ'}(G)$. The graphs $G$ with $es_{χ'}(G)=\lfloor n/2\rfloor$ are classified, and the graphs $G$ with $es_{χ'}(G)=\lfloor n/2\rfloor-1$ and $χ'(G)=Δ(G)+1$ are characterized. We establish that the odd cycles and $K_2$ are exactly the regular connected graphs with the chromatic edge stability index $1$; on the other hand, we prove that it is NP-hard to verify whether a graph $G$ has $es_{χ'}(G)=1$. We also prove that every minimum mitigating set of an $r$-regular graph $G$, where $r\ne 4$, with $es_{χ'}(G)=2$ is a matching. Furthermore, we propose a conjecture that for every graph $G$ there exists a minimum mitigating set, which is a matching, and prove that the conjecture holds for graphs $G$ with $es_{χ'}(G)\in\{1,2,\lfloor n/2\rfloor-1,\lfloor n/2\rfloor\}$, and for bipartite graphs.

math.CO↗