Pythagorean Neutrosophic Planar Graphs with an Application in Decision-Making

Graph theory has a significant impact and is crucial in the structure of many real-life situations.To simulate uncertainty and ambiguity,many extensions of graph theoretical notions were created.Planar graphs play a vital role in modelling which has the property of non-crossing edges.Although crossing edges benefit,they have some drawbacks,which paved the way for the introduction of planar graphs.The overall purpose of the study is to contribute to the conceptual development of the Pythagorean Neutrosophic graph.The basic methodology of our research is the incorporation of the analogous concepts of planar graphs in the Pythagorean Neutrosophic graphs.The significant finding of our research is the introduction of Pythagorean Neutrosophic Planar graphs,a conceptual blending of Pythagorean Neutro-sophic and Planar graphs.The idea of Pythagorean Neutrosophic multigraphs and dual graphs are also introduced to deal with the ambiguous situations.This paper investigates the Pythagorean Neutrosophic planar values,which form the edges of the Pythagorean neutrosophic graphs.The concept of Pythagorean Neutrosophic dual graphs,isomorphism,co-weak and weak isomorphism have also been explored for Pythagorean Neutrosophic planar graphs.A decision-making algorithm was proposed with a numerical illustra-tion by using the Pythagorean Neutrosophic fuzzy graph.

Power domination in planar graphs with small diameter

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graph theory. In this paper, it was shown that the power domination number of an outerplanar graph with the diameter two or a 2-connected outerplanar graph with the diameter three is precisely one. Upper bounds on the power domination number for a general planar graph with the diameter two or three were determined as an immediate consequences of results proven by Dorfling, et al. Also, an infinite family of outerplanar graphs with the diameter four having arbitrarily large power domination numbers were given.

SOME APPLICATIONS OF PLANAR GRAPH IN KNOT THEORY

The relationship between a link diagram and its corresponding planar graph is briefly reviewed. A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot. The relationship between planar graph and almost planar Seifert surface is discussed. Using planar graph, we construct an alternating amphicheiral prime knot with crossing number n for any even number n 〉 4. This gives an affirmative answer to problem 1.66（B） on Kirby＇s problem list .

A NOTE ON STRONG EMBEDDINGS OF MAXIMAL PLANAR GRAPHS ON NON ORIENTABLE SURFACES

In this paper, it is shown that for every maximal planar graph G=(V,E) , a strong embedding on some non orientable surface with genus at most |V(G)|-22 is admitted such that the surface dual of G is also a planar graph. As a corollary, an interpolation theorem for strong embeddings of G on non orientable surfaces is obtained.

Bondage Number of 1-Planar Graph

The bondage number of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph a domination number greater than the domination number of G. In this paper, we prove that for a 1-planar graph G.

Incidence Coloring of Outer-1-planar Graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices lie on the outer-face and each edge crosses at most one another edge.It is known that every outer-1-planar graph is a planar partial3-tree.In this paper,we conjecture that every planar graph G has a proper incidence(Δ(G)+2)-coloring and confirm it for outer-1-planar graphs with maximum degree at least 8 or with girth at least 4.Specifically,we prove that every outer-1-planar graph G has an incidence(Δ(G)+3,2)-coloring,and every outer-1-planar graph G with maximum degree at least 8 or with girth at least 4 has an incidence(Δ(G)+2,2)-coloring.

Improper Choosability of Planar Graphs without 6-circuits

图 G 被叫(k， d )*-choosable 如果为为所有 v V (G) 的每表任务 L 令人满意的 | L (v)|=k，有 G 的 L 着色以便 G 的每个顶点至多有与象自己的一样的颜色有颜色的 d 邻居。在这份报纸，没有 6 电路和邻近自己或四角形的一个三角形的每张平面图是，这被显示出(3,1 )*-choosable。

Edge Colorings of Planar Graphs without 6-Cycles with Two Chords

It is proved here that if a planar graph has maximum degree at least 6 and any 6-cycle contains at most one chord, then it is of class 1.

Neighbor Sum Distinguishing Total Choosability of Planar Graphs with Maximum Degree at Least 10

A neighbor sum distinguishing(NSD)total coloringφof G is a proper total coloring of G such thatΣz∈EG(u)U{u}φ(z)≠Σz∈EG(v)U{v}φ(z)for each edge uv∈E(G),where EG(u)is the set of edges incident with a vertex u.In 2015,Pilśniak and Wozniak conjectured that every graph with maximum degreeΔhas an NSD total(Δ+3)-coloring.Recently,Yang et al.proved that the conjecture holds for planar graphs withΔ≥10,and Qu et al.proved that the list version of the conjecture also holds for planar graphs withΔ≥13.In this paper,we improve their results and prove that the list version of the conjecture holds for planar graphs withΔ≥10.

Fractional Coloring Planar Graphs under Steinberg-type Conditions

A Steinberg-type conjecture on circular coloring is recently proposed that for any prime p≥5,every planar graph of girth p without cycles of length from p+1 to p(p-2)is Cp-colorable(that is,it admits a homomorphism to the odd cycle Cp).The assumption of p≥5 being prime number is necessary,and this conjecture implies a special case of Jaeger’s Conjecture that every planar graph of girth 2p-2 is Cp-colorable for prime p≥5.In this paper,combining our previous results,we show the fractional coloring version of this conjecture is true.Particularly,the p=5 case of our fractional coloring result shows that every planar graph of girth 5 without cycles of length from 6 to 15 admits a homomorphism to the Petersen graph.

Injective △+2 Coloring of Planar Graph Without Short Cycles

A coloring of graph G is an injective coloring if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The injective chromatic number χi(G) of G is the least integer k such that G has an injective k-coloring. In this paper, we prove that(1) if G is a planar graph with girth g ≥ 6 and maximum degree △ ≥ 7, then χi(G) ≤ △ + 2;(2) if G is a planar graph with △ ≥ 24 and without 3,4,7-cycles, then χi(G) ≤ △ + 2.

Extremal P_(8)-Free/P_(9)-Free Planar Graphs

An H-free graph is a graph not containing the given graph H as a subgraph.It is well known that the Turán number ex(n,H)is the maximum number of edges in an H-free graph on n vertices.Based on this definition,we define ex_(P)(n,H)to restrict the graph classes to planar graphs,that is,ex_(P)(n,H)=max{|E(G)|:G∈G,where G is a family of all H-free planar graphs on n vertices.Obviously,we have ex_(P)(n,H)=3n−6 if the graph H is not a planar graph.The study is initiated by Dowden(J Graph Theory 83:213–230,2016),who obtained some results when H is considered as C_(4)or C_(5).In this paper,we determine the values of ex_(P)(n,Pk)with k∈{8,9},where Pk is a path with k vertices.

On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number

Let G = （V, E） be a simple graph. A function f ： E → {＋1,-1} is called a signed cycle domination function （SCDF） of G if ∑e∈E（C） f（e） ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc（G） = min{∑e∈E f（e）｜ f is an SCDF of G}. This paper will characterize all maxima] planar graphs G with order n ≥ 6 and γ′sc（G） =n.

Spectral Radius of Hamiltonian Planar Graphs and Outerplanar Graphs

The spectral radius is an important parameter of a graph related to networks. A method for estimating the spectral radius of each spanning subgraph is used to prove that the spectral radius of a Hamiltonian planar graph of order n≥4 is less than or equal to 2+3n-11 and the spectral radius of the outerplanar graph of order n≥6 is less than or equal to 22+n-5, which are improvements over previous results. A direction for further study is then suggested.

Combinatorial Invariants on Planar Graphs

This paper introduces three kinds of operators on planar graphs with binary weights on edges, for which combinatorial invariants on two kinds of equivalences are found. Further, it is shown that the Jones polynomial and the bracket polynomial which are proved to be new topological invariants on knots in topology become special cases. Moreover, these invariants are a kind of generalization of Tutte polynomial on graphs.

Neighbor Sum Distinguishing Total Choice Number of Planar Graphs without 6-cycles

Pilsniak and Wozniak put forward the concept of neighbor sum distinguishing(NSD)total coloring and conjectured that any graph with maximum degreeΔadmits an NSD total(Δ+3)-coloring in 2015.In 2016,Qu et al.showed that the list version of the conjecture holds for any planar graph withΔ≥13.In this paper,we prove that any planar graph withΔ≥7 but without 6-cycles satisfies the list version of the conjecture.

Group Edge Choosability of Planar Graphs without Adjacent Short Cycles

In this paper, we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group （△ （G）＋1）-edge-choosable, and some planar graphs with large girth and maximum degree are group △（G）-edge-choosable.

Light Subgraphs in the Family of 1-Planar Graphs with High Minimum Degree

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2 V （K1 ∪ K2） with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1 V （K1∪K2） with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.

A Note on the Minimum Total Coloring of Planar Graphs

Graph coloring is an important tool in the study of optimization, computer science, network design, e.g., file transferring in a computer network, pattern matching, computation of Hessians matrix and so on. In this paper, we consider one important coloring, vertex coloring of a total graph, which is also called total coloring. We consider a planar graph G with maximum degree △（G） 〉 8, and proved that if G contains no adjacent i,j-cycles with two chords for some i,j E {5,6,7}, then G is total-（△ ＋ 1）-colorable.

Acyclic Total Colorings of Planar Graphs without l Cycles

A proper total coloring of a graph G such that there are at least 4 colors on those vertices and edges incident with a cycle of G, is called acyclic total coloring. The acyclic total chromatic number of G is the least number of colors in an acyclic total coloring of G. In this paper, it is proved that the acyclic total chromatic number of a planar graph G of maximum degree at least k and without 1 cycles is at most △（G） ＋ 2 if （k, l） ∈ {（6, 3）, （7, 4）, （6, 5）, （7, 6）}.