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article

Eigenvalues and $[a, b]$ ‐factors in regular graphs

Suil O

Journal of Graph Theory > 100 > 3 > 458 - 469

For positive integers,

r \geq 3

,

h \geq 1

, and

k \geq 1

, Bollobás, Saito, and Wormald proved some sufficient conditions for an

h

‐edge‐connected

r

‐regular graph to have a

k

‐factor in 1985. Lu gave an upper bound for the third largest eigenvalue in a connected

r

‐regular graph to have a

k

‐factor in 2010. Gu found an upper bound for certain eigenvalues in an

h

‐edge‐connected

r

‐regular graph to have a...

article

Edge colorings and circular flows on regular graphs

Davide Mattiolo, Eckhard Steffen

Journal of Graph Theory > 99 > 3 > 399 - 413

Let

ϕ_{c} (G)

be the circular flow number of a bridgeless graph

G

. By Steffen it was proved that, for a bridgeless

(2 t + 1)

‐regular graph

G

, where

t \geq 1

is a positive integer,

ϕ_{c} (G) \in {2 + \frac{1}{t}, 2 + \frac{2}{2 t - 1}}

if and only if

G

has a perfect matching

M

such that

G - M

is bipartite. This implies that

G

is a class 1 graph. For

t = 1

, all graphs with circular flow number bigger than...

article

Highly edge‐connected regular graphs without large factorizable subgraphs

Davide Mattiolo, Eckhard Steffen

Journal of Graph Theory > 99 > 1 > 107 - 116

We construct highly edge‐connected

r

‐regular graphs of even order which do not contain

r - 2

pairwise disjoint perfect matchings. When

r

is a multiple of 4, the result solves a problem of Thomassen [4].

article

Fixation properties of multiple cooperator configurations on regular graphs

Hendrik Richter

Theory in Biosciences > 2019 > 138 > 2 > 261-275

Whether or not cooperation is favored in evolutionary games on graphs depends on the population structure and spatial properties of the interaction network. The population structure can be expressed as configurations. Such configurations extend scenarios with a single cooperator among defectors to any number of cooperators and any arrangement of cooperators and defectors on the network. For interaction...

article

On edge product cordial graphs

Jaroslav Ivanco

Opuscula Mathematica > 2019 > Vol. 39, no. 5 > 691--703

An edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize graphs admitting an edge product cordial labeling. Using this characterization we investigate the edge product cordiality of broad classes of graphs, namely, dense graphs, dense bipartite graphs, connected regular graphs, unions of some graphs, direct products of some bipartite graphs, joins...

article

Elliptic semiplanes and regular graphs with girth 5

E. Abajo, C. Balbuena, M. Bendala

Electronic Notes in Discrete Mathematics > 2018 > 68 > C > 245-250

A (k, g)-graph is a k-regular graph with girth g and a (k, g)-cage is a (k, g)-graph with the fewest possible number of vertices. The cage problem consists of constructing (k, g)-graphs of minimum order n(k, g). We focus on girth g=5, where cages are known only for degrees k≤7. Considering the relationship between finite geometries and graphs we establish upper constructive bounds on n(k, 5), for...

article

The smallest singular value of a shifted d-regular random square matrix

Alexander E. Litvak, Anna Lytova, Konstantin Tikhomirov, Nicole Tomczak-Jaegermann, more

Probability Theory and Related Fields > 2019 > 173 > 3-4 > 1301-1347

We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let $$C_1<d< c n/\log ^2 n$$ C 1 < d < c n / log 2 n and let $$\mathcal {M}_{n,d}$$ M n , d be the set of all $$n\times n$$ n × n square matrices with 0 / 1 entries, such that each row...

article

Covering 2‐connected 3‐regular graphs with disjoint paths

Gexin Yu

Journal of Graph Theory > 88 > 3 > 385 - 401

A path cover of a graph is a set of disjoint paths so that every vertex in the graph is contained in one of the paths. The path cover number

p (G)

of graph G is the cardinality of a path cover with the minimum number of paths. Reed in 1996 conjectured that a 2‐connected 3‐regular graph has path cover number at most

⌈ n / 10 ⌉

. In this article, we confirm this conjecture.

article

On Uniquely k-List Colorable Planar Graphs, Graphs on Surfaces, and Regular Graphs

M. Abdolmaleki, J. P. Hutchinson, S. Gh. Ilchi, E. S. Mahmoodian, more

Graphs and Combinatorics > 2018 > 34 > 3 > 383-394

A graph G is called uniquely k -list colorable (UkLC) if there exists a list of colors on its vertices, say $$L=\lbrace S_v \mid v \in V(G) \rbrace $$ L={Sv∣v∈V(G)} , each of size k, such that there is a unique proper list coloring of G from this list of colors. A graph G is said to have property M(k) if it is not uniquely k-list colorable. Mahmoodian and Mahdian (Ars Comb 51:295–305, 1999) characterized...

article

A new method for proving dominating uniqueness of graphs

Gee Choon Lau

Journal of the Association of Arab Universities for Basic and Applied... > 2017 > 24 > C > 292-299

Let G be a graph of order n. A subset S of V(G) is a dominating set of G if every vertex in V(G)⧹S is adjacent to at least one vertex of S. The domination polynomial of G is the polynomial D(G,x)=∑i=γ(G)nd(G,i)xi, where d(G,i) is the number of dominating sets of G of size i, and γ(G) is the size of a smallest dominating set of G, called the domination number of G. We say two graphs G and H are dominating...

article

On sets of type $${\varvec{(m,m+q)_2}}$$ ( m , m + q ) 2 in $${\varvec{PG(3,q)}}$$ P G ( 3 , q )

Fulvio Zuanni

Journal of Geometry > 2017 > 108 > 3 > 1137-1155

A set K of type $$(m,m+q)_2$$ ( m , m + q ) 2 in the projective space PG(3, q) is a set of points such that every plane meets K in either m or $$m+q$$ m + q points and, furthermore, there are both planes meeting K in m points and planes meeting K in $$m+q$$ m + q points. The classical example of such a set is a partial spread of size m; however, several other families...

article

Strong-majority bootstrap percolation on regular graphs with low dissemination threshold

Dieter Mitsche, Xavier Pérez-Giménez, Paweł Prałat

Stochastic Processes and their Applications > 2017 > 127 > 9 > 3110-3134

Consider the following model of strong-majority bootstrap percolation on a graph. Let r≥1 be some integer, and p∈[0,1]. Initially, every vertex is active with probability p, independently from all other vertices. Then, at every step of the process, each vertex v of degree deg(v) becomes active if at least (deg(v)+r)/2 of its neighbours are active. Given any arbitrarily small p>0 and any integer...

article

A Note on the Locating-Total Domination in Graphs

Mirka Miller, R. Sundara Rajan, R. Jayagopal, Indra Rajasingh, more

Discussiones Mathematicae Graph Theory > 2017 > 37 > 3 > 745-754

In this paper we obtain a sharp (improved) lower bound on the locating-total domination number of a graph, and show that the decision problem for the locating-total domination is NP-complete.

article

Alliance polynomial of regular graphs

Walter Carballosa, José M. Rodríguez, José M. Sigarreta, Yadira Torres-Nuñez

Discrete Applied Mathematics > 2017 > 225 > C > 22-32

The alliance polynomial of a graph G with order n and maximum degree Δ is the polynomial A(G;x)=∑k=−ΔΔAk(G)xn+k, where Ak(G) is the number of exact defensive k-alliances in G. We obtain some properties of A(G;x) and its coefficients for regular graphs. In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove...

article

On regular handicap graphs of even order

Dalibor Fronček, Petr Kovář, Tereza Kovářová, Bohumil Krajc, more

Electronic Notes in Discrete Mathematics > 2017 > 60 > C > 69-76

Let G=(V,E) be a simple graph of order n. A bijection f : V→{1,2,…,n} is a handicap labeling of G if there exists an integer ℓ such that ∑u∈N(v)f(u)=ℓ+f(v) for all v∈V, where N(v) is the set of all vertices adjacent to v. Any graph which admits a handicap labeling is a handicap graph.We present an overview of results, which completely answer the question of existence of regular handicap graphs of...

article

Edge Weighting Functions on Semitotal Dominating Sets

Michael A. Henning

Graphs and Combinatorics > 2017 > 33 > 2 > 403-417

A set S of vertices in an isolate-free graph G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number is the minimum cardinality of a semitotal dominating set of G, and is bounded below by the domination number and bounded above by the total domination number, arguably the two most important...

article

Optimal regular graph designs

Sera Aylin Cakiroglu

Statistics and Computing > 2018 > 28 > 1 > 103-112

A typical problem in optimal design theory is finding an experimental design that is optimal with respect to some criteria in a class of designs. The most popular criteria include the A- and D-criteria. Regular graph designs occur in many optimality results, and if the number of blocks is large enough, an A-optimal (or D-optimal) design is among them (if any exist). To explore the landscape of designs...

article

Confidence Sets for the Source of a Diffusion in Regular Trees

Justin Khim, Po-Ling Loh

IEEE Transactions on Network Science and Engineering > 2017 > 4 > 1 > 27 - 40

We study the problem of identifying the source of a diffusion spreading over a regular tree. When the degree of each node is at least three, we show that it is possible to construct confidence sets for the diffusion source with size independent of the number of infected nodes. Our estimators are motivated by analogous results in the literature concerning identification of the root node in preferential...

article

On regular graphs with four distinct eigenvalues

Xueyi Huang, Qiongxiang Huang

Linear Algebra and its Applications > 2017 > 512 > C > 219-233

Let G(4,2) be the set of connected regular graphs with four distinct eigenvalues in which exactly two eigenvalues are simple, G(4,2,−1) (resp. G(4,2,0)) the set of graphs belonging to G(4,2) with −1 (resp. 0) as an eigenvalue, and G(4,≥−1) the set of connected regular graphs with four distinct eigenvalues and second least eigenvalue not less than −1. In this paper, we prove the non-existence of connected...

article

Is there any polynomial upper bound for the universal labeling of graphs?

Arash Ahadi, Ali Dehghan, Morteza Saghafian

Journal of Combinatorial Optimization > 2017 > 34 > 3 > 760-770

A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation $$\ell $$ ℓ of G for every two adjacent vertices v and u, the sum of incoming edges of v and u in the oriented graph are different from each other. The universal labeling number of a graph G is the minimum number k such that G has universal labeling from $$\{1,2,\ldots , k\}$$ { 1 ,...

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Search results

Eigenvalues and $[a, b]$ ‐factors in regular graphs

Edge colorings and circular flows on regular graphs

Highly edge‐connected regular graphs without large factorizable subgraphs

Fixation properties of multiple cooperator configurations on regular graphs

On edge product cordial graphs

Elliptic semiplanes and regular graphs with girth 5

The smallest singular value of a shifted d-regular random square matrix

Covering 2‐connected 3‐regular graphs with disjoint paths

On Uniquely k-List Colorable Planar Graphs, Graphs on Surfaces, and Regular Graphs

A new method for proving dominating uniqueness of graphs

On sets of type $${\varvec{(m,m+q)_2}}$$ ( m , m + q ) 2 in $${\varvec{PG(3,q)}}$$ P G ( 3 , q )

Strong-majority bootstrap percolation on regular graphs with low dissemination threshold

A Note on the Locating-Total Domination in Graphs

Alliance polynomial of regular graphs

On regular handicap graphs of even order

Edge Weighting Functions on Semitotal Dominating Sets

Optimal regular graph designs

Confidence Sets for the Source of a Diffusion in Regular Trees

On regular graphs with four distinct eigenvalues

Is there any polynomial upper bound for the universal labeling of graphs?

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