scholarly journals On(a,1)-Vertex-Antimagic Edge Labeling of Regular Graphs

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Martin Bača ◽  
Andrea Semaničová-Feňovčíková ◽  
Tao-Ming Wang ◽  
Guang-Hui Zhang
Keyword(s):  
Regular Graphs ◽  
Open Problems ◽  
Arithmetic Sequence ◽  
Bijective Mapping ◽  
Vertex Weight ◽  
Positive Integers ◽  
Edge Labeling

An(a,s)-vertex-antimagic edge labeling(or an(a,s)-VAElabeling, for short) ofGis a bijective mapping from the edge setE(G)of a graphGto the set of integers1,2,…,|E(G)|with the property that the vertex-weights form an arithmetic sequence starting fromaand having common differences, whereaandsare two positive integers, and the vertex-weight is the sum of the labels of all edges incident to the vertex. A graph is called(a,s)-antimagic if it admits an(a,s)-VAElabeling. In this paper, we investigate the existence of(a,1)-VAE labeling for disconnected 3-regular graphs. Also, we define and study a new concept(a,s)-vertex-antimagic edge deficiency, as an extension of(a,s)-VAE labeling, for measuring how close a graph is away from being an(a,s)-antimagic graph. Furthermore, the(a,1)-VAE deficiency of Hamiltonian regular graphs of even degree is completely determined. More open problems are mentioned in the concluding remarks.

scholarly journals Regular Graphs and Corona Graphs Based on Special Type of Labeling

2019 ◽  
Vol 9 (2) ◽  
pp. 2662-2664
Keyword(s):  
Regular Graphs ◽  
Corona Graphs ◽  
Edge Labeling

Here we consider the special type of labeling as lucky edge labeling for Regular graphs and corona graphs.

scholarly journals On the Total Vertex Irregular Labeling of Proper Interval Graphs

2020 ◽  
Vol 12 (4) ◽  
pp. 537-543
Author(s):  
A. Rana
Keyword(s):  
Efficient Algorithm ◽  
Simple Graph ◽  
Interval Graphs ◽  
Vertex Weight ◽  
Vertex Set ◽  
Positive Integers ◽  
Irregularity Strength

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers).  For a simple graph G = (V, E) with vertex set V and edge set E, a labeling  Φ: V ∪ E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x∈ V under a total k-labeling  Φ is defined as wt(x) = Φ(x) + ∑y∈N(x) Φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph, if for every two different vertices x and y of G, wt(x)≠wt(y). The minimum k for which  a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs is presented for interval graphs.

scholarly journals Super H -Antimagic Total Covering for Generalized Antiprism and Toroidal Octagonal Map

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Amir Taimur ◽  
Gohar Ali ◽  
Muhammad Numan ◽  
Adnan Aslam ◽  
Kraidi Anoh Yannick
Keyword(s):  
Arithmetic Sequence ◽  
Total Graph ◽  
Bijective Function ◽  
Positive Integers

Let G be a graph and H ⊆ G be subgraph of G . The graph G is said to be a , d - H antimagic total graph if there exists a bijective function f : V H ∪ E H ⟶ 1,2,3 , … , V H + E H such that, for all subgraphs isomorphic to H , the total H weights W H = W H = ∑ x ∈ V H f x + ∑ y ∈ E H f y forms an arithmetic sequence a , a + d , a + 2 d , … , a + n − 1 d , where a and d are positive integers and n is the number of subgraphs isomorphic to H . An a , d - H antimagic total labeling f is said to be super if the vertex labels are from the set 1,2 , … , | V G . In this paper, we discuss super a , d - C 3 -antimagic total labeling for generalized antiprism and a super a , d - C 8 -antimagic total labeling for toroidal octagonal map.

scholarly journals Good approximation and characterization of subgroups of R = Z

2001 ◽  
Vol 38 (1-4) ◽  
pp. 97-113 ◽  
Author(s):  
A. Bíró ◽  
J. M. Deshouillers ◽  
Vera T. Sós
Keyword(s):  
Continued Fraction ◽  
Irrational Number ◽  
Explicit Construction ◽  
Continued Fraction Expansion ◽  
Open Problems ◽  
Positive Integers

Let be a real irrational number and A =(xn) be a sequence of positive integers. We call A a characterizing sequence of or of the group Z mod 1 if lim n 2A n !1 k k =0 if and only if 2 Z mod 1. In the present paper we prove the existence of such characterizing sequences, also for more general subgroups of R = Z . Inthespecialcase Z mod 1 we give explicit construction of a characterizing sequence in terms of the continued fraction expansion of. Further, we also prove some results concerning the growth and gap properties of such sequences. Finally, we formulate some open problems.

scholarly journals Cartesian Products of Some Regular Graphs Admitting Antimagic Labeling for Arbitrary Sets of Real Numbers

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yi-Wu Chang ◽  
Shan-Pang Liu
Keyword(s):  
Complete Graphs ◽  
Regular Graphs ◽  
Real Numbers ◽  
Cartesian Products ◽  
Antimagic Labeling ◽  
Edge Labeling

An edge labeling of graph G with labels in A is an injection from E G to A , where E G is the edge set of G , and A is a subset of ℝ . A graph G is called ℝ -antimagic if for each subset A of ℝ with A = E G , there is an edge labeling with labels in A such that the sums of the labels assigned to edges incident to distinct vertices are different. The main result of this paper is that the Cartesian products of complete graphs (except K 1 ) and cycles are ℝ -antimagic.

scholarly journals Vertex Covering with Monochromatic Pieces of few Colours

10.37236/7469 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Marlo Eugster ◽  
Frank Mousset
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Independence Number ◽  
Vertex Cover ◽  
Constant Factor ◽  
Regular Graphs ◽  
Kneser Graph ◽  
Vertex Covering ◽  
Vertex Graph ◽  
Positive Integers

In 1995, Erdös and Gyárfás proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2\sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers $r,s$, what is the smallest number $pc_{r,s}(K_n)$ such that in every colouring of the edges of $K_n$ with $r$ colours, there exists a vertex cover of $K_n$ by $pc_{r,s}(K_n)$ monochromatic paths using altogether at most $s$ different colours?For fixed integers $r>s$ and as $n\to\infty$, we prove that $pc_{r,s}(K_n) = \Theta(n^{1/\chi})$, where $\chi=\max{\{1,2+2s-r\}}$ is the chromatic number of the Kneser graph $KG(r,r-s)$. More generally, if one replaces $K_n$ by an arbitrary $n$-vertex graph with fixed independence number $\alpha$, then we have $pc_{r,s}(G) = O(n^{1/\chi})$, where this time around $\chi$ is the chromatic number of the Kneser hypergraph $KG^{(\alpha+1)}(r,r-s)$. This result is tight in the sense that there exist graphs with independence number $\alpha$ for which $pc_{r,s}(G) = \Omega(n^{1/\chi})$. This is in sharp contrast to the case $r=s$, where it follows from a result of Sárközy (2012) that $pc_{r,r}(G)$ depends only on $r$ and $\alpha$, but not on the number of vertices.We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic $d$-regular graphs.

scholarly journals On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion

CAUCHY ◽  
2021 ◽  
Vol 7 (1) ◽  
pp. 64-72
Author(s):  
Marsidi Marsidi ◽  
Ika Hesti Agustin ◽  
Dafik Dafik ◽  
Elsa Yuli Kurniawati
Keyword(s):  
Connection Number ◽  
Vertex Weight ◽  
Edge Labeling

All graph in this paper are simple, finite, and connected. Let  be a labeling of a graph . The function  is called antimagic rainbow edge labeling if for any two vertices  and , all internal vertices in path  have different weight, where the weight of vertex is the sum of its incident edges label. The vertex weight denoted by  for every . If G has a antimagic rainbow edge labeling, then  is a antimagic rainbow vertex connection, where the every vertex is assigned with the color . The antimagic rainbow vertex connection number of , denoted by , is the minimum colors taken over all rainbow vertex connection induced by antimagic rainbow edge labeling of . In this paper, we determined the exact value of the antimagic rainbow vertex connection number of path ( ), wheel ( ), friendship ( ), and fan ( ).

scholarly journals Bounded-Degree Graphs have Arbitrarily Large Geometric Thickness

10.37236/1029 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
János Barát ◽  
Jiří Matoušek ◽  
David R. Wood
Keyword(s):  
Graph Drawing ◽  
Line Drawing ◽  
Geometric Graphs ◽  
Regular Graphs ◽  
Open Problems ◽  
Bounded Degree ◽  
Large N ◽  
Straight Line ◽  
Minimum Integer ◽  
Geometric Thickness

The geometric thickness of a graph $G$ is the minimum integer $k$ such that there is a straight line drawing of $G$ with its edge set partitioned into $k$ plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists $\Delta$-regular graphs with arbitrarily large geometric thickness. In particular, for all $\Delta\geq9$ and for all large $n$, there exists a $\Delta$-regular graph with geometric thickness at least $c\sqrt{\Delta}\,n^{1/2-4/\Delta-\epsilon}$. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th International Symp. on Graph Drawing (GD '04), vol. 3383 of Lecture Notes in Comput. Sci., Springer, 2004] and Ambrus et al. [The slope parameter of graphs. Tech. Rep. MAT-2005-07, Department of Mathematics, Technical University of Denmark, 2005].

scholarly journals Thue-like Sequences and Rainbow Arithmetic Progressions

10.37236/1660 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Jaroslaw Grytczuk
Keyword(s):  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Euclidean Spaces ◽  
Open Problems ◽  
Lovász Local Lemma ◽  
Fixed Integer ◽  
Local Lemma ◽  
Minimum Number ◽  
Lovasz Local Lemma ◽  
Positive Integers

A sequence $u=u_{1}u_{2}...u_{n}$ is said to be nonrepetitive if no two adjacent blocks of $u$ are exactly the same. For instance, the sequence $a{\bf bcbc}ba$ contains a repetition $bcbc$, while $abcacbabcbac$ is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set $\{a,b,c\}$. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this paper we consider a stronger property defined as follows. Let $k\geq 2$ be a fixed integer and let $C$ denote a set of colors (or symbols). A coloring $f:{\bf N}\rightarrow C$ of positive integers is said to be $k$-nonrepetitive if for every $r\geq 1$ each segment of $kr$ consecutive numbers contains a $k$-term rainbow arithmetic progression of difference $r$. In particular, among any $k$ consecutive blocks of the sequence $f=f(1)f(2)f(3)...$ no two are identical. By an application of the Lovász Local Lemma we show that the minimum number of colors in a $k$-nonrepetitive coloring is at most $2^{-1}e^{k(2k-1)/(k-1)^{2}}k^{2}(k-1)+1$. Clearly at least $k+1$ colors are needed but whether $O(k)$ suffices remains open. This and other types of nonrepetitiveness can be studied on other structures like graphs, lattices, Euclidean spaces, etc., as well. Unlike for the classical Thue sequences, in most of these situations non-constructive arguments seem to be unavoidable. A few of a range of open problems appearing in this area are presented at the end of the paper.

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