arXiv subjects

Manouchehr Zaker

Publications and source records attributed to Manouchehr Zaker.

At least 19 recordsLinked to original sources

arXiv2026

On ${\rm b}^{\ast}$-Coloring and $z$-Coloring of graphs with high girth

In a proper vertex coloring $c$ of a graph $G$, a vertex $u$ is called a b-vertex if $u$ is adjacent to a vertex in every other color class. A ${\rm b}^{\ast}$-coloring is a proper coloring in which a b-vertex is adjacent to a b-vertex in every other color class. A Grundy coloring is a proper coloring obtained by the First-Fit (greedy) coloring procedure. A $z$-coloring of $G$ is a ${\rm b}^{\ast}$-coloring that is also a Grundy coloring. The ${\rm b}^{\ast}$-chromatic number (resp., $z$-chromatic number), denoted by ${\rm b}^{\ast}(G)$ (resp., $z(G)$), is the maximum number of colors used in a ${\rm b}^{\ast}$-coloring (resp., $z$-coloring) of $G$. Every graph admits a ${\rm b}^{\ast}$-coloring and a $z$-coloring that can be found using a polynomial-time coloring heuristic. Let ${\rm m}^{\ast}(G)$ be the largest integer $k$ such that a vertex of degree at least $k$ in $G$ has $k$ neighbors of degree at least $k$. We employ list-coloring techniques to prove that if $G$ has a girth of at least $7$, then ${\rm b}^{\ast}(G) = {\rm m}^{\ast}(G)+ 1$. A similar result is obtained for graphs of girth at least $6$ when ${\rm m}^{\ast}=3$. Finally, we obtain some results for the $z$-chromatic number. We prove that if the girth is at least $2m^{\ast}(G)+4$ and $G$ contains a specific tree as an ordinary subgraph, then $z(G)= m^{\ast}(G)+1$.

math.CO↗

arXiv2026

Improved bounds on the b-chromatic number using the independence and chromatic numbers

A b-coloring of a graph $G$ is a proper vertex coloring where each color class contains at least one vertex (a b-vertex) adjacent to a vertex in every other color class. The maximum number of colors in such a coloring is the b-chromatic number, ${\rm b}(G)$. A ${\rm b}^{\ast}$-coloring is a variation in which a b-vertex is adjacent to a b-vertex in every other color class. We employ the ${\rm b}^{\ast}$-coloring to prove that any $n$-vertex graph $G$ with independence number at most $t$ satisfies ${\rm b}(G) \leq [(t-1)n+t\chi(G)]/(2t-1)$. This bound extends the bounds of Kouider and Zaker (2006) and Alkhateeb and Kohl (2011) and improves the bound in terms of the clique partition number. We show that this bound is sharp for all $t\geq 2$ and $\chi(G)\geq 3$. Furthermore, we provide a refined bound based on the maximum number of vertex-disjoint independent sets of size $t$. Finally, we prove ${\rm b}^{\ast}(G) \leq [(t-2)n+(t-1)\chi(G)]/(2t-3)$ for all $K_{1,t}$-free graphs $G$, a significant improvement over the analogous bound for ${\rm b}(G)$.

math.CO↗

arXiv2026

Computing welfare and fairness in allocating identical goods with entitlements and general utility functions

A number of goods are called identical if they provide the same level of utility to each agent. In various real-world instances of fair division scenarios, identical indivisible items are allocated to consumers and demandants with different entitlements. We assume that the utility of $t$ identical items to each agent $A$ equals $f_A(t)$, where $f_A$ is an arbitrary increasing function corresponding to $A$. We present a polynomial time algorithm that determines the maximum weighted Rawlsian and Leximin welfare for scenarios with identical goods and show that the allocation obtained by the algorithm is equitable up to any item (WEQX). Some results concerning restricted utilitarian welfare and existence of WMMS and WEFX allocations are also presented. We introduce a new quantity ``total weighted deficit," for allocations, and by which we obtain a tractable algorithm to achieve equitable allocations for scenarios with identical goods and different weights via compensation by using a minimum number of identical coins. Some result are for scenarios with $k$ goods of different types.

math.OC↗

arXiv2024

On z-coloring and ${\rm b}^{\ast}$-coloring of graphs as improved variants of the b-coloring

Let $G$ be a simple graph and $c$ a proper vertex coloring of $G$. A vertex $u$ is called b-vertex in $(G,c)$ if all colors except $c(u)$ appear in the neighborhood of $u$. By a ${\rm b}^{\ast}$-coloring of $G$ using colors $\{1, \ldots, k\}$ we define a proper vertex coloring $c$ such that there is a b-vertex $u$ (called nice vertex) such that for each $j\in \{1, \ldots, k\}$ with $j\not=c(u)$, $u$ is adjacent to a b-vertex of color $j$. The ${\rm b}^{\ast}$-chromatic number of $G$ (denoted by ${\rm b}^{\ast}(G)$) is the largest integer $k$ such that $G$ has a ${\rm b}^{\ast}$-coloring using $k$ colors. Every graph $G$ admits a ${\rm b}^{\ast}$-coloring which is an improvement over the famous b-coloring. A z-coloring of $G$ is a coloring $c$ using colors $\{1, 2, \ldots, k\}$ containing a nice vertex of color $k$ such that for each two colors $i<j$, each vertex of color $j$ has a neighbor of color $i$ in the graph (i.e. $c$ is obtained from a greedy coloring of $G$). We prove that ${\rm b}^{\ast}(G)$ cannot be approximated within any constant factor unless $P=NP$. We obtain results for ${\rm b}^{\ast}$-coloring and z-coloring of block graphs, cacti, $P_4$-sparse graphs and graphs with girth greater than $4$. We prove that z-coloring and ${\rm b}^{\ast}$-coloring have a locality property. A linear 0-1 programming model is also presented for z-coloring of graphs. The positive results suggest that researches can be focused on ${\rm b}^{\ast}$-coloring (or z-coloring) instead of b-coloring of graphs.

math.CO↗

arXiv2024

$\mathcal{O}(VE)$ time algorithms for the Grundy (First-Fit) chromatic number of block graphs and graphs with sufficiently large girth

The Grundy (or First-Fit) chromatic number of a graph $G=(V,E)$, denoted by $Γ(G)$ (or $χ_{_{\sf FF}}(G)$), is the maximum number of colors used by a First-Fit (greedy) coloring of $G$. To determine $Γ(G)$ is NP-complete for various classes of graphs. Also there exists a constant $c>0$ such that the Grundy number is hard to approximate within the ratio $c$. We first obtain an $\mathcal{O}(VE)$ algorithm to determine the Grundy number of block graphs i.e. graphs in which every biconnected component is complete subgraph. We prove that the Grundy number of a general graph $G$ with cut-vertices is upper bounded by the Grundy number of a block graph corresponding to $G$. This provides a reasonable upper bound for the Grundy number of graphs with cut-vertices. Next, define $Δ_2(G)={\max}_{u\in G}~ {\max}_{v\in N(u):d(v)\leq d(u)} d(v)$. We obtain an $\mathcal{O}(VE)$ algorithm to determine $Γ(G)$ for graphs $G$ whose girth $g$ is at least $2Δ_2(G)+1$. This algorithm provides a polynomial time approximation algorithm within ratio $\min \{1, (g+1)/(2Δ_2(G)+2)\}$ for $Γ(G)$ of general graphs $G$ with girth $g$.

math.CO↗

arXiv2024

More results on the $z$-chromatic number of graphs

By a $z$-coloring of a graph $G$ we mean any proper vertex coloring consisting of the color classes $C_1, \ldots, C_k$ such that $(i)$ for any two colors $i$ and $j$ with $1 \leq i < j \leq k$, any vertex of color $j$ is adjacent to a vertex of color $i$, $(ii)$ there exists a set $\{u_1, \ldots, u_k\}$ of vertices of $G$ such that $u_j \in C_j$ for any $j \in \{1, \ldots, k\}$ and $u_k$ is adjacent to $u_j$ for each $1 \leq j \leq k$ with $j \not=k$, and $(iii)$ for each $i$ and $j$ with $i \not= j$, the vertex $u_j$ has a neighbor in $C_i$. Denote by $z(G)$ the maximum number of colors used in any $z$-coloring of $G$. Denote the Grundy and {\rm b}-chromatic number of $G$ by $Γ(G)$ and ${\rm b}(G)$, respectively. The $z$-coloring is an improvement over both the Grundy and b-coloring of graphs. We prove that $z(G)$ is much better than $\min\{Γ(G), {\rm b}(G)\}$ for infinitely many graphs $G$ by obtaining an infinite sequence $\{G_n\}_{n=3}^{\infty}$ of graphs such that $z(G_n)=n$ but $Γ(G_n)={\rm b}(G_n)=2n-1$ for each $n\geq 3$. We show that acyclic graphs are $z$-monotonic and $z$-continuous. Then it is proved that to decide whether $z(G)=Δ(G)+1$ is $NP$-complete even for bipartite graphs $G$. We finally prove that to recognize graphs $G$ satisfying $z(G)=χ(G)$ is $coNP$-complete, improving a previous result for the Grundy number.

math.CO↗

arXiv2022

Bounds for the Grundy chromatic number of graphs in terms of domination number

For any graph $G$, the Grundy (or First-Fit) chromatic number of $G$, denoted by $Γ(G)$ (also $χ_{_{\sf FF}}(G)$), is defined as the maximum number of colors used by the First-Fit (greedy) coloring of the vertices of $G$. Determining the Grundy number is $NP$-complete, and obtaining bounds for $Γ(G)$ in terms of the known graph parameters is an active research topic. By a star partition of $G$ we mean any partition of $V(G)$ into say $V_1, \ldots, V_k$ such that each $G[V_i]$ contains a vertex adjacent to any other vertex in $V_i$. In this paper using the star partition of graphs we obtain the first upper bounds for the Grundy number in terms of the domination number. We also prove some bounds in terms of the domination number and girth of graphs.

math.CO↗

arXiv2021

More relations between $λ$-labeling and Hamiltonian paths with emphasis on line graph of bipartite multigraphs

This paper deals with the $λ$-labeling and $L(2,1)$-coloring of simple graphs. A $λ$-labeling of a graph $G$ is any labeling of the vertices of $G$ with different labels such that any two adjacent vertices receive labels which differ at least two. Also an $L(2,1)$-coloring of $G$ is any labeling of the vertices of $G$ such that any two adjacent vertices receive labels which differ at least two and any two vertices with distance two receive distinct labels. Assume that a partial $λ$-labeling $f$ is given in a graph $G$. A general question is whether $f$ can be extended to a $λ$-labeling of $G$. We show that the extension is feasible if and only if a Hamiltonian path consistent with some distance constraints exists in the complement of $G$. Then we consider line graph of bipartite multigraphs and determine the minimum number of labels in $L(2,1)$-coloring and $λ$-labeling of these graphs. In fact we obtain easily computable formulas for the path covering number and the maximum path of the complement of these graphs. We obtain a polynomial time algorithm which generates all Hamiltonian paths in the related graphs. A special case is the Cartesian product graph $K_n\Box K_n$ and the generation of $λ$-squares.

math.CO↗

arXiv2020

On Grundy and b-chromatic number of some families of graphs: a comparative study

The Grundy and the {\rm b}-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph $G$, denoted by $Γ(G)$ is the worst case behavior of greedy (First-Fit) coloring procedure for $G$ and the {\rm b}-chromatic number ${\rm{b}}(G)$ is the maximum number of colors used in any color-dominating coloring of $G$. Because the nature of these colorings are different they have been studied widely but separately in the literature. This paper presents a comparative study of these coloring parameters. There exists a sequence $\{G_n\}_{n\geq 1}$ with limited {\rm b}-chromatic number but $Γ(G_n)\rightarrow \infty$. We obtain families of graphs $\mathcal{F}$ such that for some adequate function $f(.)$, $Γ(G)\leq f({\rm{b}}(G))$, for each graph $G$ from the family. This verifies a previous conjecture for these families.

math.CO↗

arXiv2020

A new vertex coloring heuristic and corresponding chromatic number

One method to obtain a proper vertex coloring of graphs using a reasonable number of colors is to start from any arbitrary proper coloring and then repeat some local re-coloring techniques to reduce the number of color classes. The Grundy (First-Fit) coloring and color-dominating colorings of graphs are two well-known such techniques. The color-dominating colorings are also known and commonly referred as {\rm b}-colorings. But these two topics have been studied separately in graph theory. We introduce a new coloring procedure which combines the strategies of these two techniques and satisfies an additional property. We first prove that the vertices of every graph $G$ can be effectively colored using color classes say $C_1, \ldots, C_k$ such that $(i)$ for any two colors $i$ and $j$ with $1\leq i< j \leq k$, any vertex of color $j$ is adjacent to a vertex of color $i$, $(ii)$ there exists a set $\{u_1, \ldots, u_k\}$ of vertices of $G$ such that $u_j\in C_j$ for any $j\in \{1, \ldots, k\}$ and $u_k$ is adjacent to $u_j$ for each $1\leq j \leq k$ with $j\not= k$, and $(iii)$ for each $i$ and $j$ with $i\not= j$, the vertex $u_j$ has a neighbor in $C_i$. This provides a new vertex coloring heuristic which improves both Grundy and color-dominating colorings. Denote by $z(G)$ the maximum number of colors used in any proper vertex coloring satisfying the above properties. The $z(G)$ quantifies the worst-case behavior of the heuristic. We prove the existence of $\{G_n\}_{n\geq 1}$ such that $\min \{Γ(G_n), b(G_n)\} \rightarrow \infty$ but $z(G_n)\leq 3$ for each $n$. For each positive integer $t$ we construct a family of finitely many colored graphs ${\mathcal{D}}_t$ satisfying the property that if $z(G)\geq t$ for a graph $G$ then $G$ contains an element from ${\mathcal{D}}_t$ as a colored subgraph. This provides an algorithmic method for proving numeric upper bounds for $z(G)$.

cs.DM↗

arXiv2020

A note concerning the Grundy and ${\rm b}$-chromatic number of graphs

The Grundy number of a graph $G$ is the maximum number of colors used by the First-Fit coloring of $G$ and is denoted by $Γ(G)$. Similarly, the ${\rm b}$-chromatic number ${\rm{b}}(G)$ of $G$ expresses the worst case behavior of another well-known coloring procedure i.e. color-dominating coloring of $G$. We obtain some families of graphs $\mathcal{F}$ for which there exists a function $f(x)$ such that $Γ(G)\leq f({\rm{b}}(G))$, for each graph $G$ from the family. Call any such family $(Γ,b)$-bounded family. We conjecture that the family of ${\rm b}$-monotone graphs is $(Γ,b)$-bounded and validate the conjecture for some families of graphs.

math.CO↗

arXiv2019

On irreversible spread of influence in edge-weighted graphs

Various kinds of spread of influence occur in real world social and virtual networks. These phenomena are formulated by activation processes and irreversible dynamic monopolies in combinatorial graphs representing the topology of the networks. In most cases the nature of influence is weighted and the spread of influence depends on the weight of edges. The ordinary formulation and results for dynamic monopolies do not work for such models. In this paper we present a graph theoretical analysis for spread of weighted influence and mention a real world example realizing the activation model with weighted influence. Then we obtain some extremal bounds and algorithmic results for activation process and dynamic monopolies in directed and undirected graphs with weighted edges.

cs.DM↗

arXiv2019

A memory theoretic approach for investigating the roles of language and intuition in mathematical thinking activities

Questions concerning origin of mathematical knowledge and roles of language and intuition (imagery) in mathematical thoughts are long standing and widely debated. By introspection, mathematicians usually have some beliefs regarding these questions. But these beliefs are usually in a big contrast with the recent cognitive theoretic findings concerning mathematics. Contemporary cognitive science opens new approaches to reformulate the fundamental questions concerning mathematics and helps mathematicians break through the Platonic beliefs about the essence and sources of mathematical knowledge. In this article, we introduce and discuss mathematical thinking activities and fundamental processes such as symbolic/formal and visual/spatial ones. Two different aspects of mathematics should be separated in mathematical cognition. One aspect considers mathematics as an explicit crystallized knowledge. The other aspect considers mathematics as an ongoing and transient mental processing. The cognitive processes and corresponding tasks involved in these aspects are different. Ongoing mathematical activities both elementary and advanced, demand working memory resources. Using dual-task techniques, we design some pilot experiments to differentiate the symbolic/formal and visual/spatial processes. Using this memory theoretic approach, we explain the crucial roles of language-based processes such as verbal articulation and instructive speech and also visuo-spatial intuition such as spatial imagery and mental movement in various aspects of mathematics.

q-bio.NC↗

arXiv2018

Partial vertex covers and the complexity of some problems concerning static and dynamic monopolies

Let $G$ be a graph and $τ$ be an assignment of nonnegative integer thresholds to the vertices of $G$. Denote the average of thresholds in $τ$ by $\barτ$. A subset of vertices $D$ is said to be a $τ$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$ and for any $i\in \{0, \ldots, k-1\}$, each vertex $v$ in $D_{i+1}$ has at least $τ(v)$ neighbors in $D_0\cup \ldots \cup D_i$. Denote the size of smallest $τ$-dynamic monopoly by $dyn_τ(G)$. Also a subset of vertices $M$ is said to be a $τ$-static monopoly (or simply $τ$-monopoly) if any vertex $v\in V(G)\setminus M$ has at least $τ(v)$ neighbors in $M$. Denote the size of smallest $τ$-monopoly by $mon_τ(G)$. For a given positive number $t$, denote by $Sdyn_t(G)$ (resp. $Smon_t(G)$), the minimum $dyn_τ(G)$ (resp. $mon_τ(G)$) among all threshold assignments $τ$ with $\overlineτ\geq t$. In this paper we consider the concept of partial vertex cover as follows. Let $G=(V, E)$ be a graph and $t$ be any positive integer. A subset $S\subseteq V$ is said to be a $t$-partial vertex cover of $G$, if $S$ covers at least $t$ edges of $G$. Denote the smallest size of a $t$-partial vertex cover of $G$ by $Pβ_t(G)$. Let $ρ$, $0<ρ<1$ be any fixed number and $G$ be a given bipartite graph with $m$ edges. We first prove that to determine the smallest cardinality of a set $S\subseteq V(G)$ such that $S$ covers at least $ρm$ edges of $G$, is an NP-hard problem. Then we prove that for any constant $t$, $Sdyn_{t}(G)=Pβ_{nt-m}(G)$ and $Smon_t(G)=Pβ_{nt/2}(G)$, where $n$ and $m$ are the order and size of $G$, respectively.

math.CO↗

arXiv2018

Weak dynamic monopolies in social graphs

Dynamic monopolies were already defined and studied for the formulation of the phenomena of the spread of influence in social networks such as disease, opinion, adaptation of new product and etc. The elements of the network which have been influenced (e.g. infected or adapted an opinion) are called active vertices. It is assumed in these models that when an element is activated, it remains active until the end of the process. But in some phenomena of the spread of influence this property does not hold. For example in some diseases the infection lasts only a limited period of time or consider the spread of disease or propagation of computer virus together with some quarantination or decontamination methods. Dynamic monopolies are not useful for the study of these latter phenomena. For this purpose, we introduce a new model for such diffusions of influence and call it weak dynamic monopoly. A social network is represented by a graph $G$. Assume that any vertex $v$ of $G$ has a threshold $τ(v)\in \Bbb{N}$. Then a subset $D\subseteq V(G)$ is said to be a weak dynamic monopoly if $V(G)$ can be partitioned into $D_0=D, D_1, \ldots, D_k$ such that for any $i$, any vertex $v$ of $D_i$ has at least $τ(v)$ neighbors in $D_{i-1}$. In this definition, by the size and the processing time of $D$ we mean $|D|$ and $k$, respectively. We first investigate the relationships between weak dynamic monopolies and other related concepts and then obtain some bounds for the smallest size of weak dynamic monopolies. Next we obtain some results concerning the processing time of weak dynamic monopolies in terms of some graph parameters. Finally, a hardness result concerning inapproximibility of the determining the smallest size of weak dynamic monopolies in general graphs is obtained.

math.CO↗

arXiv2017

On monopoly and dynamic monopoly of Cartesian product of graphs with constant thresholds

Let $G$ be any simple and undirected graph. By a threshold assignment $τ$ in $G$ we mean any function $τ:V(G)\rightarrow \mathbb{N}$ such that $τ(v)\leq d_G(v)$ for any vertex $v$ of $G$. Given a graph $G$ with a threshold assignment $τ$, a subset of vertices $M$ is said to be a $τ$-monopoly if there exist at least $τ(v)$ neighbors in $M$ for any vertex $v \in V(G) \setminus M$. Similarly, a subset of vertices $D$ is said to be a $τ$-dynamic monopoly if starting with the set $D$ and iteratively adding to the current set further vertices $u$ that have at least $τ(u)$ neighbors in it, results in the entire vertex set $V(G)$. Denote by $mon_τ(G)$ (resp. $dyn_τ(G)$) the smallest cardinality of a $τ$-monopoly (resp. $τ$-dynamic monopoly) of the graph among all others. In this paper we obtain some lower and upper bounds for these two parameters with constant threshold assignments for Cartesian product graphs. Our bounds improve the previous known bounds. We also determine the exact value of these two parameters with fixed thresholds in some Cartesian graph products including cycles and complete graphs.

math.CO↗

arXiv2016

First-Fit coloring of Cartesian product graphs and its defining sets

Let the vertices of a Cartesian product graph $G\Box H$ be ordered by an ordering $σ$. By the First-Fit coloring of $(G\Box H, σ)$ we mean the vertex coloring procedure which scans the vertices according to the ordering $σ$ and for each vertex assigns the smallest available color. Let $FF(G\Box H,σ)$ be the number of colors used in this coloring. By introducing the concept of descent we obtain a sufficient condition to determine whether $FF(G\Box H,σ)=FF(G\Box H,τ)$, where $σ$ and $τ$ are arbitrary orders. We study and obtain some bounds for $FF(G\Box H,σ)$, where $σ$ is any quasi-lexicographic ordering. The First-Fit coloring of $(G\Box H, σ)$ does not always yield an optimum coloring. A greedy defining set of $(G\Box H, σ)$ is a subset $S$ of vertices in the graph together with a suitable pre-coloring of $S$ such that by fixing the colors of $S$ the First-Fit coloring of $(G\Box H, σ)$ yields an optimum coloring. We show that the First-Fit coloring and greedy defining sets of $G\Box H$ with respect to any quasi-lexicographic ordering (including the known lexicographic order) are all the same. We obtain upper and lower bounds for the smallest cardinality of a greedy defining set in $G\Box H$, including some extremal results for Latin squares.

math.CO↗