scholarly journals Zero-Sum Magic Labelings and Null Sets of Regular Graphs

10.37236/3654 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Saieed Akbari ◽  
Farhad Rahmati ◽  
Sanaz Zare
Keyword(s):  
Bipartite Graph ◽  
Regular Graph ◽  
Regular Graphs ◽  
Natural Numbers ◽  
Vertex Set ◽  
Zero Sum ◽  
Null Set

For every $h\in  \mathbb{N}$, a graph $G$ with the vertex set $V(G)$ and the  edge set $E(G)$ is said to be $h$-magic if there exists a labeling $l : E(G) \rightarrow\mathbb{Z}_h \setminus \{0\}$  such that the induced vertex labeling $s : V (G) \rightarrow \mathbb{Z}_h$, defined  by $s(v) =\sum_{uv \in E(G)} l(uv)$ is a constant map. When this constant is zero, we say that $G$ admits a zero-sum $h$-magic labeling. The null set of a graph $G$, denoted by $N(G)$, is the set of all natural numbers $h \in \mathbb{ N} $ such that $G$ admits a zero-sum $h$-magic labeling.  In 2012, the null sets of 3-regular graphs were determined. In this paper we show that if $G$ is an $r$-regular graph, then for even $r$ ($r > 2$), $N(G)=\mathbb{N}$ and for odd $r$ ($r\neq5$),  $\mathbb{N} \setminus \{2,4\}\subseteq N(G)$. Moreover,  we prove that if $r$ is odd and $G$ is a $2$-edge connected $r$-regular graph ($r\neq 5$), then $ N(G)=\mathbb{N} \setminus \{2\}$. Also, we show that if $G$ is a $2$-edge connected bipartite graph, then $\mathbb{N} \setminus \{2,3,4,5\}\subseteq N(G)$.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite 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.

scholarly journals Binary Representations of Regular Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
Yury J. Ionin
Keyword(s):  
Regular Graph ◽  
Line Graph ◽  
Strongly Regular Graph ◽  
Regular Graphs ◽  
Spherical Representation ◽  
Least Eigenvalue ◽  
Vertex Set ◽  
Strongly Regular ◽  
Distance Set ◽  
Hamming Space

For any 2-distance set in the n-dimensional binary Hamming space , let be the graph with as the vertex set and with two vertices adjacent if and only if the distance between them is the smaller of the two nonzero distances in . The binary spherical representation number of a graph , or bsr(), is the least n such that is isomorphic to , where is a 2-distance set lying on a sphere in . It is shown that if is a connected regular graph, then bsr, where b is the order of and m is the multiplicity of the least eigenvalue of , and the case of equality is characterized. In particular, if is a connected strongly regular graph, then bsr if and only if is the block graph of a quasisymmetric 2-design. It is also shown that if a connected regular graph is cospectral with a line graph and has the same binary spherical representation number as this line graph, then it is a line graph.

scholarly journals A Prolific Construction of Strongly Regular Graphs with the $n$-e.c. Property

10.37236/1647 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Dudley Stark
Keyword(s):  
Regular Graph ◽  
Prime Power ◽  
Probabilistic Methods ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Existentially Closed ◽  
Large N ◽  
Vertex Set ◽  
Strongly Regular ◽  
Hadamard Designs

A graph is $n$-e.c.$\,$ ($n$-existentially closed) if for every pair of subsets $U$, $W$ of the vertex set $V$ of the graph such that $U\cap W=\emptyset$ and $|U|+|W|=n$, there is a vertex $v\in V-(U\cup W)$ such that all edges between $v$ and $U$ are present and no edges between $v$ and $W$ are present. A graph is strongly regular if it is a regular graph such that the number of vertices mutually adjacent to a pair of vertices $v_1,v_2\in V$ depends only on whether or not $\{v_1,v_2\}$ is an edge in the graph. The only strongly regular graphs that are known to be $n$-e.c. for large $n$ are the Paley graphs. Recently D. G. Fon-Der-Flaass has found prolific constructions of strongly regular graphs using affine designs. He notes that some of these constructions were also studied by Wallis. By taking the affine designs to be Hadamard designs obtained from Paley tournaments, we use probabilistic methods to show that many non-isomorphic strongly regular $n$-e.c. graphs of order $(q+1)^2$ exist whenever $q\geq 16 n^2 2^{2n}$ is a prime power such that $q\equiv 3\!\!\!\pmod{4}$.

scholarly journals Perfect Matchings in $\epsilon$-regular Graphs

10.37236/1351 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Noga Alon ◽  
Vojtech Rödl ◽  
Andrzej Ruciński
Keyword(s):  
Bipartite Graph ◽  
Regular Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Perfect Matchings ◽  
Regular Graphs

A super $(d,\epsilon)$-regular graph on $2n$ vertices is a bipartite graph on the classes of vertices $V_1$ and $V_2$, where $|V_1|=|V_2|=n$, in which the minimum degree and the maximum degree are between $ (d-\epsilon)n$ and $ (d+\epsilon) n$, and for every $U \subset V_1, W \subset V_2$ with $|U| \geq \epsilon n$, $|W| \geq \epsilon n$, $|{{e(U,W) }\over{|U||W|}}-{{e(V_1,V_2)}\over{|V_1||V_2|}}| < \epsilon.$ We prove that for every $1>d >2 \epsilon >0$ and $n>n_0(\epsilon)$, the number of perfect matchings in any such graph is at least $(d-2\epsilon)^n n!$ and at most $(d+2 \epsilon)^n n!$. The proof relies on the validity of two well known conjectures for permanents; the Minc conjecture, proved by Brégman, and the van der Waerden conjecture, proved by Falikman and Egorichev.

scholarly journals Random Regular Graphs: Asymptotic Distributions and Contiguity

1995 ◽  
Vol 4 (4) ◽  
pp. 369-405 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Regular Graph ◽  
Asymptotic Distributions ◽  
Regular Graphs ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Vertex Set ◽  
Multiple Edges ◽  
The One ◽  
Unusual Type ◽  
Infinite Linear

The asymptotic distribution of the number of Hamilton cycles in a random regular graph is determined. The limit distribution is of an unusual type; it is the distribution of a variable whose logarithm can be written as an infinite linear combination of independent Poisson variables, and thus the logarithm has an infinitely divisible distribution with a certain discrete Lévy measure. Similar results are found for some related problems. These limit results imply that some different models of random regular graphs are contiguous, which means that they are qualitatively asymptotically equivalent. For example, if r > 3, then the usual (uniformly distributed) random r-regular graph is contiguous to the one constructed by taking the union of r perfect matchings on the same vertex set (assumed to be of even cardinality), conditioned on there being no multiple edges. Some consequences of contiguity for asymptotic distributions are discussed.

scholarly journals A Note on Zero-Sum 5-Flows in Regular Graphs

10.37236/2145 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Saieed Akbari ◽  
Narges Ghareghani ◽  
Gholamreza Khosrovshahi ◽  
Sanaz Zare
Keyword(s):  
Natural Number ◽  
Regular Graph ◽  
Affirmative Answer ◽  
Regular Graphs ◽  
Real Numbers ◽  
Zero Sum

Let $G$ be a graph. A zero-sum flow of $G$ is an assignment of non-zero real numbers to the edges of $G$ such that the sum of the values of all edges incident with each vertex is zero. Let $k$ be a natural number. A zero-sum $k$-flow is a flow with values from the set $\{\pm 1,\ldots ,\pm(k-1)\}$. It has been conjectured that every $r$-regular graph, $r\geq 3$, admits a zero-sum $5$-flow. In this paper we provide an affirmative answer to this conjecture, except for  $r=5$.

The maximum number of vertices of primitive regular graphs of orders 2, 3, 4 with exponent 2

2021 ◽  
pp. 97-104
Author(s):  
M. B. Abrosimov ◽  
◽  
S. V. Kostin ◽  
I. V. Los ◽  
◽  
...  
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Similar Question

In 2015, the results were obtained for the maximum number of vertices nk in regular graphs of a given order k with a diameter 2: n2 = 5, n3 = 10, n4 = 15. In this paper, we investigate a similar question about the largest number of vertices npk in a primitive regular graph of order k with exponent 2. All primitive regular graphs with exponent 2, except for the complete one, also have diameter d = 2. The following values were obtained for primitive regular graphs with exponent 2: np2 = 3, np3 = 4, np4 = 11.

Minimal Regular Graphs of Girths Eight and Twelve

1966 ◽  
Vol 18 ◽  
pp. 1091-1094 ◽  
Author(s):  
Clark T. Benson
Keyword(s):  
Group Theory ◽  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

HAMILTONIAN PROPERTIES OF FAULTY RECURSIVE CIRCULANT GRAPHS

2002 ◽  
Vol 03 (03n04) ◽  
pp. 273-289 ◽  
Author(s):  
CHANG-HSIUNG TSAI ◽  
JIMMY J. M. TAN ◽  
YEN-CHU CHUANG ◽  
LIH-HSING HSU
Keyword(s):  
Bipartite Graph ◽  
Hamiltonian Path ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Vertex Set ◽  
Regular Node ◽  
Hamiltonian Properties

We present some results concerning hamiltonian properties of recursive circulant graphs in the presence of faulty vertices and/or edges. The recursive circulant graph G(N, d) with d ≥ 2 has vertex set V(G) = {0, 1, …, N - 1} and the edge set E(G) = {(v, w)| ∃ i, 0 ≤ i ≤ ⌈ log d N⌉ - 1, such that v = w + di (mod N)}. When N = cdk where d ≥ 2 and 2 ≤ c ≤ d, G(cdk, d) is regular, node symmetric and can be recursively constructed. G(cdk, d) is a bipartite graph if and only if c is even and d is odd. Let F, the faulty set, be a subset of V(G(cdk, d)) ∪ E(G(cdk, d)). In this paper, we prove that G(cdk, d) - F remains hamiltonian if |F| ≤ deg (G(cdk, d)) - 2 and G(cdk, d) is not bipartite. Moreover, if |F| ≤ deg (G(cdk, d)) - 3 and G(cdk, d) is not a bipartite graph, we prove a more stronger result that for any two vertices u and v in V(G(cdk, d)) - F, there exists a hamiltonian path of G(cdk, d) - F joining u and v.

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