Locally-iterative Distributed (Δ + 1)-coloring and Applications

We consider graph coloring and related problems in the distributed message-passing model. Locally-iterative algorithms are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1-hop-neighborhood . In STOC’93 Szegedy and Vishwanathan showed that any locally-iterative Δ + 1-coloring algorithm requires Ω (Δ log Δ + log * n ) rounds, unless there exists “a very special type of coloring that can be very efficiently reduced” [ 44 ]. No such special coloring has been found since then. This led researchers to believe that Szegedy-Vishwanathan barrier is an inherent limitation for locally-iterative algorithms and to explore other approaches to the coloring problem [ 2 , 3 , 19 , 32 ]. The latter gave rise to faster algorithms, but their heavy machinery that is of non-locally-iterative nature made them far less suitable to various settings. In this article, we obtain the aforementioned special type of coloring. Specifically, we devise a locally-iterative Δ + 1-coloring algorithm with running time O (Δ + log * n ), i.e., below Szegedy-Vishwanathan barrier. This demonstrates that this barrier is not an inherent limitation for locally-iterative algorithms. As a result, we also achieve significant improvements for dynamic, self-stabilizing, and bandwidth-restricted settings. This includes the following results: We obtain self-stabilizing distributed algorithms for Δ + 1-vertex-coloring, (2Δ - 1)-edge-coloring, maximal independent set, and maximal matching with O (Δ + log * n ) time. This significantly improves previously known results that have O(n) or larger running times [ 23 ]. We devise a (2Δ - 1)-edge-coloring algorithm in the CONGEST model with O (Δ + log * n ) time and O (Δ)-edge-coloring in the Bit-Round model with O (Δ + log n ) time. The factors of log * n and log n are unavoidable in the CONGEST and Bit-Round models, respectively. Previously known algorithms had superlinear dependency on Δ for (2Δ - 1)-edge-coloring in these models. We obtain an arbdefective coloring algorithm with running time O (√ Δ + log * n ). Such a coloring is not necessarily proper, but has certain helpful properties. We employ it to compute a proper (1 + ε)Δ-coloring within O (√ Δ + log * n ) time and Δ + 1-coloring within O (√ Δ log Δ log * Δ + log * n ) time. This improves the recent state-of-the-art bounds of Barenboim from PODC’15 [ 2 ] and Fraigniaud et al. from FOCS’16 [ 19 ] by polylogarithmic factors. Our algorithms are applicable to the SET-LOCAL model [ 25 ] (also known as the weak LOCAL model). In this model a relatively strong lower bound of Ω (Δ 1/3 ) is known for Δ + 1-coloring. However, most of the coloring algorithms do not work in this model. (In Reference [ 25 ] only Linial’s O (Δ 2 )-time algorithm and Kuhn-Wattenhofer O (Δ log Δ)-time algorithms are shown to work in it.) We obtain the first linear-in-Δ Δ + 1-coloring algorithms that work also in this model.

The majority coloring of the join and Cartesian product of some digraph

A majority coloring of a digraph is a vertex coloring such that for every vertex, the number of vertices with the same color in the out-neighborhood does not exceed half of its out-degree. Kreutzer, Oum, Seymour and van der Zyper proved that every digraph is majority 4-colorable and conjecture that every digraph has a majority 3-coloring. This paper mainly studies the majority coloring of the joint and Cartesian product of some special digraphs and proved the conjecture is true for the join graph and the Cartesian product. According to the influence of the number of vertices in digraph, we prove the majority coloring of the joint and Cartesian product of some digraph.

On local antimagic vertex coloring of corona products related to friendship and fan graph

Let <em>G</em>=(<em>V</em>,<em>E</em>) be connected graph. A bijection <em>f </em>: <em>E</em> → {1,2,3,..., |<em>E</em>|} is a local antimagic of <em>G</em> if any adjacent vertices <em>u,v</em> ∈ <em>V</em> satisfies <em>w</em>(<em>u</em>)≠ <em>w</em>(<em>v</em>), where <em>w</em>(<em>u</em>)=∑_e∈E(u)<em>f</em>(<em>e</em>), <em>E</em>(<em>u</em>) is the set of edges incident to <em>u</em>. When vertex <em>u</em> is assigned the color <em>w</em>(<em>u</em>), we called it a local antimagic vertex coloring of <em>G</em>. A local antimagic chromatic number of <em>G</em>, denoted by <em>χ</em>_la(<em>G</em>), is the minimum number of colors taken over all colorings induced by the local antimagic labeling of <em>G</em>. In this paper, we determine the local antimagic chromatic number of corona product of friendship and fan with null graph on <em>m</em> vertices, namely, <em>χ</em>_la(<em>F</em>_n ⊙ \overline{K_m}) and <em>χ</em>_la(<em>f</em>_(1,n) ⊙ \overline{K_m}).

Graph Entropy Based on Strong Coloring of Uniform Hypergraphs

The classical graph entropy based on the vertex coloring proposed by Mowshowitz depends on a graph. In fact, a hypergraph, as a generalization of a graph, can express complex and high-order relations such that it is often used to model complex systems. Being different from the classical graph entropy, we extend this concept to a hypergraph. Then, we define the graph entropy based on the vertex strong coloring of a hypergraph. Moreover, some tightly upper and lower bounds of such graph entropies as well as the corresponding extremal hypergraphs are obtained.

On the Packing Partitioning Problem on Directed Graphs

This work is aimed to continue studying the packing sets of digraphs via the perspective of partitioning the vertex set of a digraph into packing sets (which can be interpreted as a type of vertex coloring of digraphs) and focused on finding the minimum cardinality among all packing partitions for a given digraph D, called the packing partition number of D. Some lower and upper bounds on this parameter are proven, and their exact values for directed trees are given in this paper. In the case of directed trees, the proof results in a polynomial-time algorithm for finding a packing partition of minimum cardinality. We also consider this parameter in digraph products. In particular, a complete solution to this case is presented when dealing with the rooted products.

PEWARNAAN TITIK TOTAL SUPER ANTI-AJAIB LOKAL PADA GRAF PETERSEN DIPERUMUM P(n,k) DENGAN k=1,2

The local antimagic total vertex labeling of graph G is a labeling that every vertices and edges label by natural number from 1 to such that every two adjacent vertices has different weights, where is The sum of a vertex label and the labels of all edges that incident to the vertex. If the labeling start the smallest label from the vertex then the edge so that kind of coloring is called the local super antimagic total vertex labeling. That local super antimagic total vertex labeling induces vertex coloring of graph G where for vertex v, the weight w(v) is the color of v. The minimum number of colors that obtained by coloring that induces by local super antimagic total vertex labeling of G called the chromatic number of local super antimagic total vertex coloring of G, denoted by χlsat(G). In this paper, we consider the chromatic number of local super antimagic total vertex coloring of Generalized Petersen Graph P(n,k) for k=1, 2.

The Local Antimagic Chromatic Numbers of Some Join Graphs

Let G=(V(G),E(G)) be a connected graph with n vertices and m edges. A bijection f:E(G)→{1,2,⋯,m} is an edge labeling of G. For any vertex x of G, we define ω(x)=∑e∈E(x)f(e) as the vertex label or weight of x, where E(x) is the set of edges incident to x, and f is called a local antimagic labeling of G, if ω(u)≠ω(v) for any two adjacent vertices u,v∈V(G). It is clear that any local antimagic labelling of G induces a proper vertex coloring of G by assigning the vertex label ω(x) to any vertex x of G. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of different vertex labels taken over all colorings induced by local antimagic labelings of G. In this paper, we present explicit local antimagic chromatic numbers of Fn∨K2¯ and Fn−v, where Fn is the friendship graph with n triangles and v is any vertex of Fn. Moreover, we explicitly construct an infinite class of connected graphs G such that χla(G)=χla(G∨K2¯), where G∨K2¯ is the join graph of G and the complement graph of complete graph K2. This fact leads to a counterexample to a theorem of Arumugam et al. in 2017, and our result also provides a partial solution to Problem 3.19 in Lau et al. in 2021.

Bilangan Kromatik Pewarnaan Titik pada Graf Dual dari Graf Roda

Penelitian ini bertujuan mengkonstruksi graf dual dari graf roda (Wn*) dan menentukan bilangan kromatik graf dual dari graf roda (Wn*). Penelitian ini dimulai dari menggambarkan beberapa graf roda dari ke , kemudian membangun graf dual dari graf roda dengan memanfaatkan graf-graf dari ke , kemudian memberikan warna pada titik-titik dari graf dualnya dengan menentukan bilangan kromatiknya. Diperoleh hasil bahwa Graf roda merupakan graf self-dual karena isomorfik dengan graf dualnya yaitu . Pewarnaan titik diperoleh dengan menentukan bilangan kromatik graf dual dari graf roda, menentukan pola dari bilangan kromatik, dan memberikan warna. Berdasarkan hasil penelitian, diperoleh bilangan kromatik pewarnaan titik pada graf dual dari graf roda yakni Kata Kunci: Pewarnaan Titik, Bilangan Kromatik, Graf Dual dan Graf Roda.This research aims to construct a dual graph from a wheel graph (Wn*) and determine the dual graph chromatic number of the wheel graph (Wn*). This research starts from describing some wheel graph from to , then construct a dual graph from a wheel graph from to , then gives color to the vertices of the dual graph by determining the chromatic number. The result showed that the wheel graph is a self-dual graph because it is isomorphic with its dual graph, namely . The vertex coloring is obtained by determining the chromatic number of the dual graph of the wheel graph, determining the pattern of the chromatic number and giving the color. Based on the research results, the chromatic number of vertex coloring on dual graph of a wheel graph is: Keywords: Vertex Coloring, Chromatic Number, Dual Graph and Wheel Graph.

On the local irregularity vertex coloring of volcano, broom, parachute, double broom and complete multipartite graphs

Let [Formula: see text] be a simple, finite, undirected, and connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bijection [Formula: see text] is label function [Formula: see text] if [Formula: see text] and for any two adjacent vertices [Formula: see text] and [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is set ofvertices adjacent to [Formula: see text]. [Formula: see text] is called local irregularity vertex coloring. The minimum cardinality of local irregularity vertex coloring of [Formula: see text] is called chromatic number local irregular denoted by [Formula: see text]. In this paper, we verify the exact values of volcano, broom, parachute, double broom and complete multipartite graphs.

Exact 2-Distance b-Coloring and Exact 2-Distance b-Continuity of Helm Graph 𝑯𝒏

An exact 2-distance coloring of a graph 𝐺 is a coloring of vertices of 𝐺 such that any two vertices which are at distance exactly 2 receive distinct colors. An exact 2-distance chromatic number𝑒2(𝐺) of 𝐺 is the minimum 𝑘 for which 𝐺 admits an exact 2-distance coloring with 𝑘 colors. A 𝑏-coloring of 𝐺 by 𝑘 colors is a proper 𝑘-vertex coloring such that in each color class, there exists a vertex called a color dominating vertex which has a neighbor in every other color class. A vertex that has a 2-neighbor in all other color classes is called an exact 2-distance color dominating vertex (or an 𝑒2-cdv). Exact 2-distance 𝑏-coloring (or an 𝑒2𝑏-coloring) of 𝐺 is an exact 2-distance coloring such that each color class contains an 𝑒2- cdv. An exact 2-distance 𝑏-chromatic number (or an 𝑒2𝑏-number) 𝑒2𝑏(𝐺) of 𝐺 is the largest integer 𝑘 such that 𝐺 has an 𝑒2𝑏-coloring with 𝑘colors. If for each integer𝑘, 𝑒2(𝐺) ≤ 𝑘 ≤ 𝑒2𝑏(𝐺), 𝐺 has an 𝑒2𝑏-coloring by 𝑘 colors, then 𝐺 is said to be an exact 2-distance 𝑏- continuous graph. In this paper, the 𝑒2𝑏-number𝑒2𝑏(𝐻𝑛)of the helm graph 𝐻𝑛is obtained and 𝑒2𝑏-continuity of 𝐻𝑛is discussed.