We define biequivalent planar graphs, which are a generalisation of the uniform polyhedron graphs, as planar graphs made out of two families of equivalent nodes. Such graphs are required to identify polyhedral cages with geometries suitable for artif...
Graph theory plays a substantial role in structuring and designing many problems. A number of structural designs with crossings can be found in real world scenarios. To model the vagueness and uncertainty in graphical network problems, many extension...
The Total Coloring Conjecture (TCC) states that every simple graph G is totally (Δ+2)-colorable, where Δ denotes the maximum degree of G. In this paper, we prove that TCC holds for dumbbell maximal planar graphs. Especially, we divide the...
Many symmetric properties are well-explored in graph theory, especially in graph coloring, such as symmetric graphs defined by the automorphism groups, symmetric drawing of planar graphs, and symmetric functions which are used to count the number of...
The acyclic chromatic index χa′(G) of a graph G is the smallest k for which G is a proper edge colorable using k colors. A 1-planar graph is a graph that can be drawn in plane such that every edge is crossed by at most one other edge. In th...
The field of graph theory plays a vital role in various fields. One of the important areas in graph theory is graph labeling used in many applications such as coding theory, X-ray crystallography, radar, astronomy, circuit design, communication netwo...
Let k and n be two positive integers. Firefighting is a discrete dynamical process of preventing the spread of fire. Let G be a connected graph G with n vertices. Assuming a fire starts at one of the vertices of G, the firefighters choose k unburned...
The edge DP-chromatic number of G, denoted by χDP′(G), is the minimum k such that G is edge DP-k-colorable. In 1999, Juvan, Mohar, and Thomas proved that the edge list chromatic number of K4-minor free graph G with Δ≥3 is Δ....
An equitable k-coloring of a graph G is a proper vertex coloring such that the size of any two color classes differ at most 1. If there is an equitable k-coloring of G, then the graph G is said to be equitably k-colorable. A 1-planar graph is a graph...
Symmetry, such as structural symmetry, color symmetry and so on, plays an important role in graph coloring. In this paper, we use structural symmetry and color symmetry to study the characterization for the neighbor-distinguishing index of planar gra...
It has been shown that there is a planar graph without 3-cycles which is not (Δ1,Δ2)-colorable for any given Δ1,Δ2. This inspires many research to obtain sufficient conditions for planar graphs without 4-cycles and other cycle...
Given a finite, simple, connected graph G with at least one edge, the independent bondage number bi(G) of G is the minimum size of an edge set, such that its deletion results in a graph with a strictly larger independent domination number than that o...
In simple graphs, DP-coloring is a generalization of list coloring and thus many results of DP-coloring generalize those of list coloring. Xu and Wu proved that every planar graph without 5-cycles adjacent simultaneously to 3-cycles and 4-cycles is 4...
In order to resolve Borodin’s Conjecture, DP-coloring was introduced in 2017 to extend the concept of list coloring. In previous works, it is proved that every planar graph without 7-cycles and butterflies is DP-4-colorable. And any planar grap...
A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪...
Face routing has been adopted in wireless sensor networks (WSNs) where topological changes occur frequently or maintaining full network information is difficult. For message forwarding in networks, a planar graph is used to prevent looping, and becau...
In this paper, we show that if a planar graph G satisfies the following conditions: (i) none of its 3-faces is adjacent to a 6−-face, and (ii) none of its 4-faces is adjacent to a 5−-face, then V(G) can be partitioned into two subsets, ea...
A proper total k-coloring ϕ of G with ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each uv∈E(G) is called a total neighbor sum distinguishing k-coloring, where EG(u)={uv|uv∈E(G)}. Pilśniak and Woźniak conjectured that every graph with maximum degree Δ exist...
DP-coloring as a generalization of list coloring was introduced by Dvořák and Postle recently. In this paper, we prove that every planar graph in which the distance between 6−-cycles is at least 2 is DP-3-colorable, which extends the res...
A defectivek-coloring is a coloring on the vertices of a graph with colors 1,2,…,k where adjacent vertices may have the same color. A (d1,d2)-coloring of a graph G is a defective two-coloring such that each vertex colored by i has at most di a...
Given a signed graph (G,σ) and a positive real number r, if there exists a vertex mapping c:V(G)→[0,r) satisfying that for every positive edge wx, 1≤|c(w)−c(x)|≤r−1 and for every negative edge wx, |c(w)−c(x)|≤r2&...
A neighbor full sum distinguishing total coloring of a graph G is a proper k-total coloring ϕ such that no two adjacent vertices u and v satisfy ω(u)≠ω(v), where ω(v)=ϕ(v)+∑e∋vϕ(e)+∑u∈N(v)ϕ(u)...
This paper introduces and studies the following beyond-planarity problem, which we call h-Clique2Path Planarity. Let G be a simple topological graph whose vertices are partitioned into subsets of size at most h, each inducing a clique. h-Clique2Path...
DP-coloring was introduced by Dvořák and Postle [J. Comb. Theory Ser. B 2018, 129, 38–54]. In this paper, we prove that every planar graph in which the 5−-cycles are at distance of at least 3 from each other is DP-3-colorable, whic...
Symmetry in non-planar graphs is a fundamental concept that enhances understanding, simplifies analyses, and has practical implications in diverse fields such as science, engineering, and mathematics. A total κ-labeling for a graph Gνˇ...
The dominating set problem is a core NP-hard problem in combinatorial optimization and graph theory, and has many important applications. Baker [JACM 41,1994] introduces a k-outer planar graph decomposition-based framework for designing polynomial ti...
A subgraph H of an edge-colored graph G is called rainbow if all of its edges have different colors. Let ar(G,H) denote the maximum positive integer t, such that there is a t-edge-colored graph G without any rainbow subgraph H. We denote by kK2 a mat...
In this paper, we present a new distributed approximation algorithm for the minimum dominating set problem in planar triangle-free graphs. The algorithm operates in a constant number of rounds in the LOCAL model. Using the bunch technique, we prove t...
For a commutative ring R with unity, the associate ring graph, denoted by AG(R), is a simple graph with vertices as nonzero elements of R and two distinct vertices are adjacent if they are associates. The graph AG(R) contains components equal in numb...
In this paper, the use of the construction of random processes on graphs allows us to expand the models of the theory of queuing and reliability by constructing. These problems are important because the emphasis on the legal component largely determi...
Experimentally observed complex networks are often scale-free, small-world and have an unexpectedly large number of small cycles. An Apollonian network is one notable example of a model network simultaneously having all three of these properties. Thi...
We show that the so-called motorcycle graph of a planar polygon can be constructed by a randomized incremental algorithm that is simple and experimentally fast. Various test data are given, and a clustering method for speeding up the construction is...
This article examines quantum group symmetry using the Potts model. The transformation of the Potts model into a polynomial knot state on Kaufman square brackets is analyzed. It is shown how a dichromatic polynomial for a planar graph can be obtained...
Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics...
State estimation and mapping based on Light Detection and Ranging (LIDAR) are important for autonomous systems. Point cloud registration is a crucial module affecting the accuracy and real-time performance of LIDAR simultaneous localization and mappi...
In econophysics, the achievements of information filtering methods over the past 20 years, such as the minimal spanning tree (MST) by Mantegna and the planar maximally filtered graph (PMFG) by Tumminello et al., should be celebrated. Here, we show ho...
Relatedness is a key concept in economic complexity, since the assessment of the similarity between industrial sectors enables policymakers to design optimal development strategies. However, among the different ways to quantify relatedness, a measure...
Point clouds from laser scanners have been widely used in recent research on indoor modeling methods. Currently, particularly in data-driven modeling methods, data preprocessing for dividing structural components and nonstructural components is requi...
Let S(m,e) be a class of numerical semigroups with multiplicity m and embedding dimension e. We call a graph GS an S(m,e)-graph if there exists a numerical semigroup S∈S(m,e) with V(GS)={x:x∈g(S)} and E(GS)={xy⇔x+y∈S}, where g(S)...
Let Λ denote a commutative ring with unity and D(Λ) denote a collection of all annihilating ideals from Λ. An annihilator intersection graph of Λ is represented by the notation AIG(Λ). This graph is not directed in...
We analyze the dimension of a financial correlation-based network and apply our analysis to characterize the complexity of the network. First, we generalize the volume-based dimension and find that it is well defined by the correlation-based network....
In this article, we present a simple method for network visualization. The proposed method is based on distmesh [P.O. Persson and G. Strang, A simple mesh generator in MATLAB, SIAM Review 46 (2004) pp. 329–345], which is a simple unstructured t...
Phased array radars are able to provide highly accurate airplane surveillance and tracking performance if they are properly calibrated. However, the ambient temperature variation and device aging could greatly deteriorate their performance. Currently...
We present a new combinatorial approach to the Ising model incorporating arbitrary bond weights on planar graphs. In contrast to existing methodologies, the exact free energy is expressed as the determinant of a set of ordered and disordered operator...
This paper studies the maximum-clique independence problem and some variations of the clique transversal problem such as the {k}-clique, maximum-clique, minus clique, signed clique, and k-fold clique transversal problems from algorithmic aspects for...
A branch of graph theory that makes use of a molecular graph is called chemical graph theory. Chemical graph theory is used to depict a chemical molecule. A graph is connected if there is an edge between every pair of vertices. A topological index is...
In this paper, an improved routing algorithm suitable for planar networks—static Zigbee and mesh networks included—is shown. The algorithm is based on the cycle description of the graph, and on a new graph model based on arrow description...
Every three-connected simple planar graph is a polyhedral graph and a cubic polyhedral graph with pentagonal and hexagonal faces is called as a classical fullerene. The aim of this paper is to survey some results about the symmetry group of cubic pol...
Based on the analysis of grooves of regular and partially regular microreliefs formed on flat surfaces, the relationship between the geometric parameters of the grooves of their microreliefs, which ensures their regularity, was revealed. The function...
One of the primary objectives of truss structure design optimization is to minimize the total weight by determining the optimal sizes of the truss members while ensuring structural stability and integrity against external loads. Trusses consist of pi...
of 3