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 The distinguishing number of Cartesian products of complete graphs

2008 ◽  
Vol 29 (4) ◽  
pp. 922-929 ◽  
Author(s):  
Wilfried Imrich ◽  
Janja Jerebic ◽  
Sandi Klavžar
Keyword(s):  
Complete Graphs ◽  
Cartesian Products ◽  
Distinguishing Number

scholarly journals Pretty Good State Transfer on Circulant Graphs

10.37236/6388 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Hiranmoy Pal ◽  
Bikash Bhattacharjya
Keyword(s):  
Complex Number ◽  
Adjacency Matrix ◽  
Transition Matrix ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Real Numbers ◽  
Cartesian Products ◽  
Edge Disjoint ◽  
State Transfer ◽  
Good State

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)}$, where $t\in {\mathbb R}$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and $v$ if there exists a sequence of real numbers $\{t_k\}$ and a complex number $\gamma$ of unit modulus such that $\lim\limits_{k\rightarrow\infty} H(t_k) e_u=\gamma e_v.$ We find that the cycle $C_n$ as well as its complement $\overline{C}_n$ admit pretty good state transfer if and only if $n$ is a power of two, and it occurs between every pair of antipodal vertices. In addition, we look for pretty good state transfer in more general circulant graphs. We prove that union (edge disjoint) of an integral circulant graph with a cycle, each on $2^k$ $(k\geq 3)$ vertices, admits pretty good state transfer. The complement of such union also admits pretty good state transfer. Using Cartesian products, we find some non-circulant graphs admitting pretty good state transfer.

scholarly journals Can Colour-Blind Distinguish Colour Palettes?

10.37236/2319 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Large Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Comprehensive Information ◽  
Lovasz Local Lemma ◽  
Delta G ◽  
Irregular Graphs

Let $c:E(G)\rightarrow [k]$ be  a colouring, not necessarily proper, of edges of a graph $G$. For a vertex $v\in V$, let $\overline{c}(v)=(a_1,\ldots,a_k)$, where $ a_i =|\{u:uv\in E(G),\;c(uv)=i\}|$, for $i\in [k].$ If we re-order the sequence $\overline{c}(v)$ non-decreasingly, we obtain a sequence $c^*(v)=(d_1,\ldots,d_k)$, called a palette of a vertex $v$. This can be viewed as the most comprehensive information about colours incident with $v$ which can be delivered by a person who is unable to name colours but distinguishes one from another. The smallest $k$ such that $c^*$ is a proper colouring of vertices of $G$ is called the colour-blind index of a graph $G$, and is denoted by dal$(G)$. We conjecture that there is a constant $K$ such that dal$(G)\leq K$ for every graph $G$ for which the parameter is well defined. As our main result we prove that $K\leq 6$ for regular graphs of sufficiently large degree, and for irregular graphs with $\delta (G)$ and $\Delta(G)$ satisfying certain conditions. The proofs are based on the Lopsided Lovász Local Lemma. We also show that $K=3$ for all regular bipartite graphs, and for complete graphs of order $n\geq 8$.

Proper Coloring Distance in Edge-Colored Cartesian Products of Complete Graphs and Cycles

2019 ◽  
Vol 29 (04) ◽  
pp. 1950016
Author(s):  
Ajay Arora ◽  
Eddie Cheng ◽  
Colton Magnant
Keyword(s):  
Connected Graph ◽  
The Other ◽  
Complete Graphs ◽  
Specific Class ◽  
Proper Coloring ◽  
Cartesian Products ◽  
General Graph ◽  
Proper Distance ◽  
Cyclic Graphs

An path that is edge-colored is called proper if no two consecutive edges receive the same color. A general graph that is edge-colored is called properly connected if, for every pair of vertices in the graph, there exists a properly colored path from one to the other. Given two vertices u and v in a properly connected graph G, the proper distance is the length of the shortest properly colored path from u to v. By considering a specific class of colorings that are properly connected for Cartesian products of complete and cyclic graphs, we present results on the proper distance between all pairs of vertices in the graph.

Perfect 1-factorizations

2019 ◽  
Vol 69 (3) ◽  
pp. 479-496 ◽  
Author(s):  
Alexander Rosa
Keyword(s):  
Pairwise Disjoint ◽  
Hamiltonian Cycle ◽  
Perfect Matching ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Cubic Graphs ◽  
Vertex Set ◽  
Early Survey

AbstractLetGbe a graph with vertex-setV=V(G) and edge-setE=E(G). A 1-factorofG(also calledperfect matching) is a factor ofGof degree 1, that is, a set of pairwise disjoint edges which partitionsV. A 1-factorizationofGis a partition of its edge-setEinto 1-factors. For a graphGto have a 1-factor, |V(G)| must be even, and for a graphGto admit a 1-factorization,Gmust be regular of degreer, 1 ≤r≤ |V| − 1.One can find in the literature at least two extensive surveys [69] and [89] and also a whole book [90] devoted to 1-factorizations of (mainly) complete graphs.A 1-factorization ofGis said to beperfectif the union of any two of its distinct 1-factors is a Hamiltonian cycle ofG. An early survey on perfect 1-factorizations (abbreviated as P1F) of complete graphs is [83]. In the book [90] a whole chapter (Chapter 16) is devoted to perfect 1-factorizations of complete graphs.It is the purpose of this article to present what is known to-date on P1Fs, not only of complete graphs but also of other regular graphs, primarily cubic graphs.

Construction of Antimagic Labeling for the Cartesian Product of Regular Graphs

2011 ◽  
Vol 5 (1) ◽  
pp. 81-87 ◽  
Author(s):  
Oudone Phanalasy ◽  
Mirka Miller ◽  
Costas S. Iliopoulos ◽  
Solon P. Pissis ◽  
Elaheh Vaezpour
Keyword(s):  
Cartesian Product ◽  
Regular Graphs ◽  
Antimagic Labeling

scholarly journals Antimagic Labeling of Regular Graphs

2015 ◽  
Vol 82 (4) ◽  
pp. 339-349 ◽  
Author(s):  
Feihuang Chang ◽  
Yu-Chang Liang ◽  
Zhishi Pan ◽  
Xuding Zhu
Keyword(s):  
Regular Graphs ◽  
Antimagic Labeling

scholarly journals Distinct Distances in Graph Drawings

10.37236/831 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Paz Carmi ◽  
Vida Dujmović ◽  
Pat Morin ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Combinatorial Geometry ◽  
Complete Graphs ◽  
Bounded Degree ◽  
Cartesian Products ◽  
Bounded Treewidth ◽  
Line Drawings ◽  
Straight Line ◽  
Minimum Number ◽  
Complete Bipartite Graphs

The distance-number of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in ${\cal O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth $2$ and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree $5$ and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$.

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 0-Sum and 1-Sum Flows in Regular Graphs

10.37236/4986 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
S. Akbari ◽  
M. Kano ◽  
S. Zare
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Natural Numbers ◽  
Real Numbers ◽  
Even Order

Let $G$ be a graph. Assume that $l$ and $k$ are two natural numbers. An $l$-sum flow on a graph $G$ is an assignment of non-zero real numbers to the edges of $G$ such that for every vertex $v$ of $G$ the sum of values of all edges incidence with $v$ equals $l$. An $l$-sum $k$-flow is an $l$-sum flow with values from the set $\{\pm 1,\ldots ,\pm(k-1)\}$. Recently, it was proved that for every $r, r\geq 3$, $r\neq 5$, every $r$-regular graph admits a $0$-sum $5$-flow. In this paper we settle a conjecture by showing that every $5$-regular graph admits a $0$-sum $5$-flow. Moreover, we prove that every $r$-regular graph of even order admits a $1$-sum $5$-flow.

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