scholarly journals Rectilinear Monotone r-Regular Planar Graphs for r = {3, 4, 5}

d'CARTESIAN ◽  
2015 ◽  
Vol 4 (1) ◽  
pp. 103
Author(s):  
Arthur Wulur ◽  
Benny Pinontoan ◽  
Mans Mananohas
Keyword(s):  
Planar Graph ◽  
Connected Graph ◽  
Planar Graphs ◽  
Infinite Family ◽  
Regular Graphs ◽  
Vertex Set ◽  
Simple Connected Graph

A graph G consists of non-empty set of vertex/vertices (also called node/nodes) and the set of lines connecting two vertices called edge/edges. The vertex set of a graph G is denoted by V(G) and the edge set is denoted by E(G). A Rectilinear Monotone r-Regular Planar Graph is a simple connected graph that consists of vertices with same degree and horizontal or diagonal straight edges without vertical edges and edges crossing. This research shows that there are infinite family of rectilinear monotone r-regular planar graphs for r = 3and r = 4. For r = 5, there are two drawings of rectilinear monotone r-regular planar graphs with 12 vertices and 16 vertices. Keywords: Monotone Drawings, Planar Graphs, Rectilinear Graphs, Regular Graphs

scholarly journals A Note on Full and Complete Binary Planar Graphs

2020 ◽  
Vol 3 (2) ◽  
pp. 70
Author(s):  
Emily L Casinillo ◽  
Leomarich F Casinillo
Keyword(s):  
Planar Graph ◽  
Connected Graph ◽  
Binary Tree ◽  
Planar Graphs ◽  
Tree Graph ◽  
Complete Binary Tree ◽  
Combinatorial Formula ◽  
Acyclic Graph ◽  
Tree Graphs ◽  
Vertex Set

<p>Let G=(V(G), E(G)) be a connected graph where V(G) is a finite nonempty set called vertex-set of G, and  E(G) is a set of unordered pairs {u, v} of distinct elements from  V(G) called the edge-set of G. If  is a connected acyclic graph or a connected graph with no cycles, then it is called a tree graph. A binary tree Tl with l levels is complete if all levels except possibly the last are completely full, and the last level has all its nodes to the left side. If we form a path on each level of a full and complete binary tree, then the graph is now called full and complete binary planar graph and it is denoted as Bn, where n is the level of the graph. This paper introduced a new planar graph which is derived from binary tree graphs. In addition, a combinatorial formula for counting its vertices, faces, and edges that depends on the level of the graph was developed.</p>

scholarly journals Combinatorial vs. Algebraic Characterizations of Completely Pseudo-Regular Codes

10.37236/309 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
M. Cámara ◽  
J. Fàbrega ◽  
M. A. Fiol ◽  
E. Garriga
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Regular Graphs ◽  
Completely Regular ◽  
Completely Regular Code ◽  
Positive Eigenvector ◽  
Vertex Set ◽  
Regular Codes ◽  
Simple Connected Graph

Given a simple connected graph $\Gamma$ and a subset of its vertices $C$, the pseudo-distance-regularity around $C$ generalizes, for not necessarily regular graphs, the notion of completely regular code. We then say that $C$ is a completely pseudo-regular code. Up to now, most of the characterizations of pseudo-distance-regularity has been derived from a combinatorial definition. In this paper we propose an algebraic (Terwilliger-like) approach to this notion, showing its equivalence with the combinatorial one. This allows us to give new proofs of known results, and also to obtain new characterizations which do not depend on the so-called $C$-spectrum of $\Gamma$, but only on the positive eigenvector of its adjacency matrix. Along the way, we also obtain some new results relating the local spectra of a vertex set and its antipodal. As a consequence of our study, we obtain a new characterization of a completely regular code $C$, in terms of the number of walks in $\Gamma$ with an endvertex in $C$.

scholarly journals The maximum Wiener index of maximal planar graphs

2020 ◽  
Vol 40 (4) ◽  
pp. 1121-1135
Author(s):  
Debarun Ghosh ◽  
Ervin Győri ◽  
Addisu Paulos ◽  
Nika Salia ◽  
Oscar Zamora
Keyword(s):  
Planar Graph ◽  
Connected Graph ◽  
Planar Graphs ◽  
Wiener Index ◽  
Maximal Planar Graph

Abstract The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an n-vertex maximal planar graph is at most $$\lfloor \frac{1}{18}(n^3+3n^2)\rfloor $$ ⌊ 1 18 ( n 3 + 3 n 2 ) ⌋ . We prove this conjecture and determine the unique n-vertex maximal planar graph attaining this maximum, for every $$ n\ge 10$$ n ≥ 10 .

A new graph radio k-coloring algorithm

2019 ◽  
Vol 11 (01) ◽  
pp. 1950005 ◽  
Author(s):  
Laxman Saha ◽  
Pratima Panigrahi
Keyword(s):  
Connected Graph ◽  
Channel Assignment ◽  
Time Complexity ◽  
Radio Spectrum ◽  
Circulant Graph ◽  
Communication Services ◽  
Channel Assignment Problem ◽  
Vertex Set ◽  
General Graphs ◽  
Simple Connected Graph

Due to the rapid growth in the use of wireless communication services and the corresponding scarcity and the high cost of radio spectrum bandwidth, Channel assignment problem (CAP) is becoming highly important. Radio [Formula: see text]-coloring of graphs is a variation of CAP. For a positive integer [Formula: see text], a radio [Formula: see text]-coloring of a simple connected graph [Formula: see text] is a mapping [Formula: see text] from the vertex set [Formula: see text] to the set [Formula: see text] of non-negative integers such that [Formula: see text] for each pair of distinct vertices [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the distance between [Formula: see text] and [Formula: see text] in [Formula: see text]. The span of a radio [Formula: see text]-coloring [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text] and the radio[Formula: see text]-chromatic number of [Formula: see text], denoted by [Formula: see text], is [Formula: see text] where the minimum is taken over all radio [Formula: see text]-coloring of [Formula: see text]. In this paper, we present two radio [Formula: see text]-coloring algorithms for general graphs which will produce radio [Formula: see text]-colorings with their spans. For an [Formula: see text]-vertex simple connected graph the time complexity of the both algorithm is of [Formula: see text]. Implementing these algorithms we get the exact value of [Formula: see text] for several graphs (for example, [Formula: see text], [Formula: see text], [Formula: see text], some circulant graph etc.) and many values of [Formula: see text], especially for [Formula: see text].

scholarly journals On the Ger\v{s}gorin disks of distance matrices of graphs

2021 ◽  
Vol 37 ◽  
pp. 709-717
Author(s):  
Mustapha Aouchiche ◽  
Bilal A. Rather ◽  
Issmail El Hallaoui
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Infinite Family ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Negative Answer ◽  
Diagonal Matrix ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian ◽  
Simple Connected Graph

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)=Tr(G)-D(G)$, and $D^{Q}(G)=Tr(G)+D(G)$ be the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix, and the distance signless Laplacian matrix of $G$, respectively. Atik and Panigrahi [2] suggested the study of the problem: Whether all eigenvalues, except the spectral radius, of $ D(G) $ and $ D^{Q}(G) $ lie in the smallest Ger\v{s}gorin disk? In this paper, we provide a negative answer by constructing an infinite family of counterexamples.

scholarly journals Near-Colorings: Non-Colorable Graphs and NP-Completeness

10.37236/3509 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
M. Montassier ◽  
P. Ochem
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Fixed Integer ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

A graph $G$ is $(d_1,...,d_l)$-colorable if the vertex set of $G$ can be partitioned into subsets $V_1,\ldots ,V_l$ such that the graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1 \leq i \leq l$. In this paper, we focus on complexity aspects of such colorings when $l=2,3$. More precisely, we prove that, for any fixed integers $k,j,g$ with $(k,j) \neq (0,0)$ and $g\geq3$, either every planar graph with girth at least $g$ is $(k,j)$-colorable or it is NP-complete to determine whether a planar graph with girth at least $g$ is $(k,j)$-colorable. Also, for any fixed integer $k$, it is NP-complete to determine whether a planar graph that is either $(0,0,0)$-colorable or non-$(k,k,1)$-colorable is $(0,0,0)$-colorable. Additionally, we exhibit non-$(3,1)$-colorable planar graphs with girth 5 and non-$(2,0)$-colorable planar graphs with girth 7. 

scholarly journals Zagreb, multiplicative zagreb indices and coindices of 〖nc〗_n (k) and 〖ca〗_3 (c_6 ) graphs

2018 ◽  
Vol 36 (2) ◽  
pp. 9-15
Author(s):  
Vida Ahmadi ◽  
Mohammad Reza Darafshe
Keyword(s):  
Connected Graph ◽  
Zagreb Indices ◽  
The Third ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Simple Connected Graph

Let  be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are defind, respectivly by: ,   and   where  is the degree of vertex u in G and uv is an edge of G, connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices of graph  are defind by:  and . The first and second Zagreb coindices of graph are defind by:  and .  and , named as multiplicative Zagreb coindices. In this article, we compute the first, second and the third Zagreb indices and the first and second multiplicative Zagreb indices of some  graphs. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.

scholarly journals The Size of Edge-critical Uniquely 3-Colorable Planar Graphs

10.37236/3228 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Naoki Matsumoto
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Planar Graphs ◽  
Infinite Family ◽  
The Other ◽  
Other Hand ◽  
Colorable Graph

A graph $G$ is uniquely $k$-colorable if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable graph for any edge $e\in E(G)$. In this paper, we prove that if $G$ is an edge-critical uniquely $3$-colorable planar graph, then $|E(G)|\leq \frac{8}{3}|V(G)|-\frac{17}{3}$. On the other hand, there exists an infinite family of edge-critical uniquely 3-colorable planar graphs with $n$ vertices and $\frac{9}{4}n-6$ edges. Our result gives a first non-trivial upper bound for $|E(G)|$.

scholarly journals An Update on Reconfiguring $10$-Colorings of Planar Graphs

10.37236/9391 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Zdeněk Dvořák ◽  
Carl Feghali
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Vertex Set

The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph~$G$ has as vertex set the set of all possible proper $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex. A result of Bousquet and Perarnau (2016) regarding graphs of bounded degeneracy implies that if $G$ is a planar graph with $n$ vertices, then $R_{12}(G)$ has diameter at most $6n$. We improve on the number of colors, showing that $R_{10}(G)$ has diameter at most $8n$ for every planar graph $G$ with $n$ vertices.

scholarly journals Partitioning planar graphs with girth at least $ 9 $ into an edgeless graph and a graph with bounded size components

2021 ◽  
Vol 1 (3) ◽  
pp. 136-144
Author(s):  
Chunyu Tian ◽  
◽  
Lei Sun
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Vertex Set ◽  
Bounded Size

<abstract><p>In this paper, we study the problem of partitioning the vertex set of a planar graph with girth restriction into parts, also referred to as color classes, such that each part induces a graph with components of bounded order. An ($ \mathcal{I} $, $ \mathcal{O}_{k} $)-partition of a graph $ G $ is the partition of $ V(G) $ into two non-empty subsets $ V_{1} $ and $ V_{2} $, such that $ G[V_{1}] $ is an edgeless graph and $ G[V_{2}] $ is a graph with components of order at most $ k $. We prove that every planar graph with girth 9 and without intersecting $ 9 $-face admits an ($ \mathcal{I} $, $ \mathcal{O}_{6} $)-partition. This improves a result of Choi, Dross and Ochem (2020) which says every planar graph with girth at least $ 9 $ admits an ($ \mathcal{I} $, $ \mathcal{O}_{9} $)-partition.</p></abstract>

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