Mathematical combinatorics : Vol. 3, 2018 / : international book series.

2018·1.8 MB·English

by  Mao, Linfan

ISBN 978-1-59973-582-5
VOLUME 3, 2018
MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
THE MADIS OF CHINESE ACADEMY OF SCIENCES AND
ACADEMY OF MATHEMATICAL COMBINATORICS & APPLICATIONS, USA
September, 2018
Vol.3, 2018 ISBN 978-1-59973-582-5
MATHEMATICAL COMBINATORICS
(INTERNATIONAL BOOK SERIES)
Edited By Linfan MAO
(www.mathcombin.com)
The Madis of Chinese Academy of Sciences and
Academy of Mathematical Combinatorics & Applications, USA
September, 2018
Aims and Scope: The Mathematical Combinatorics (International Book Series) is
a fully refereed international book series with ISBN number on each issue, sponsored by the
MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 110-160
pages approx. per volume, which publishes original research papers and survey articles in all
aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics,
non-euclidean geometry and topology and their applications to other sciences. Topics in detail
to be covered are:
Smarandache multi-spaces with applications to other sciences, such as those of algebraic
multi-systems, multi-metric spaces, , etc.. Smarandache geometries;
···
Topological graphs; Algebraic graphs; Random graphs; Combinatorial maps; Graph and
map enumeration; Combinatorial designs; Combinatorial enumeration;
Differential Geometry; Geometry on manifolds; Low Dimensional Topology; Differential
Topology; Topology of Manifolds; Geometrical aspects of Mathematical Physics and Relations
with Manifold Topology;
Applications of Smarandache multi-spaces to theoretical physics; Applications of Combi-
natorics to mathematics and theoretical physics; Mathematical theory on gravitational fields;
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combinatorics.
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Editorial Board (4th)
Editor-in-Chief
Linfan MAO Shaofei Du
ChineseAcademyofMathematicsandSystem Capital Normal University, P.R.China
Science, P.R.China Email: [email protected]
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Xiaodong Hu
Academy of Mathematical Combinatorics &
ChineseAcademyofMathematicsandSystem
Applications, USA
Science, P.R.China
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Email: [email protected]
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Yuanqiu Huang
Hunan Normal University, P.R.China
Guohua Song Email: [email protected]
Beijing University of Civil Engineering and
H.Iseri
Architecture, P.R.China
Mansfield University, USA
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Nankai University, P.R.China
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Junliang Cai Ion Patrascu
Beijing Normal University, P.R.China Fratii Buzesti National College
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Yanxun Chang Han Ren
Beijing Jiaotong University, P.R.China East China Normal University, P.R.China
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Beijing University of Civil Engineering and Politechnica University of Bucharest
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ii MathematicalCombinatorics(InternationalBookSeries)
Mingyao Xu
Peking University, P.R.China
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ChineseAcademyofMathematicsandSystem
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Y. Zhang
Department of Computer Science
Georgia State University, Atlanta, USA
Famous Words:
Mathematics, rightly viewed, posses not only truth, but supreme beauty – a
beauty cold and austere8, like that of sculpture.
By Bertrand Russell, a British philosopher, logician, mathematician.
Math.Combin.Book Ser. Vol.3(2018), 01-10
Generalized abc-Block Edge Transformation Graph Qabc(G)
When abc = +0
−
K.G.Mirajkar, Pooja B.
(DepartmentofMathematics,KarnatakUniversity’sKarnatakArtsCollege,Dharwad-580001,Karnataka,India)
Shreekant Patil
(DepartmentofMathematics,KarnatakUniversity,Dharwad-580003,Karnataka,India)
Email:[email protected],[email protected],[email protected]
Abstract: Thegeneralizedabc-blockedgetransformationgraphQ+0−(G)isagraphwhose
vertex set is the union of the edges and blocks of G, in which two vertices are adjacent
whenevercorrespondingedgesofGareadjacent oronecorrespondstoanedgeandotherto
a block of G are not incident with each other. In this paper, we study the girth, covering
invariants and the domination number of Q+0−(G). We present necessary and sufficient
conditions for Q+0−(G) to be planar, outerplanar, minimally nonouterplanar and maximal
outerplanar. Further, we establish a necessary and sufficient condition for the generalized
abc-block edge transformation graph Q+0−(G) havecrossing numberone.
KeyWords: Linegraph,abc-blockedgetransformation,generalizedabc-blockedgetrans-
formation graph, Smarandachely block-edgeH-graph.
AMS(2010): 05C10, 05C40.
§1. Introduction
Throughout the paper, we only consider simple graphs without isolated vertices. Definitions
notgivenheremaybe foundin[5]. Acut vertex ofaconnectedgraphisthe onewhoseremoval
increases the number of components. A nonseparable graph is connected, nontrivial and has
no cut vertices. A block of a graph is a maximal nonseparable subgraph. Let G=(V,E) be a
graph with block set U(G)= B ; B is a block of G . If a block B U(G) with the edge set
i i
{ } ∈
e ,e , ,e ;m 1 ,then we saythat the edge e andblockB areincidentwith eachother,
1 2 m i
{ ··· ≥ }
where 1 i m. The girth of a graph G, denoted by g(G) , is the length of the shortest cycle
≤ ≤
if any in G. Let x ( x ) denote the least (greatest) integer greater (less) than or equal to x.
⌈ ⌉ ⌊ ⌋
A vertex and an edge are said to cover eachother if they are incident. A set of vertices in
a graphG is avertex covering set,whichcoversallthe edgesofG. The vertex covering number
α (G) of G is the minimum number of vertices in a vertex covering set of G. A set of edges
0
in a graph G is an edge covering set, which covers all vertices of G. The edge covering number
1Supported by UGC-SAP DRS-III, New Delhi, India for 2016-2021: F.510/3/DRS-III/2016(SAP-I) Dated:
29th Feb. 2016.
2ReceivedDecember23,2017,Accepted May8,2018.
2 K.G.Mirajkar,PoojaB.andShreekantPatil
α (G) of G is the minimum number of edges in an edge covering set of G. A set of vertices in
1
a graph G is independent if no two of them are adjacent. The maximum number of vertices in
such a set is called the vertex independence number of G and is denoted by β (G). The set of
0
edges in a graph G is independent if no two of them are adjacent. The maximum number of
edges in such a set is called the edge independence number of G and is denoted by β (G).
1
Theline graph L(G)ofagraphGisthegraphwithvertexsetastheedgesetofGandtwo
vertices of L(G) are adjacent whenever the correspondingedges in G have a vertex in common
[5]. Theplick graph P(G)ofagraphGis the graphwhosesetofverticesisthe unionofthe set
ofedgesandblocksofGandinwhichtwoverticesareadjacentifandonlyifthecorresponding
edgesofGareadjacentoroneiscorrespondstoanedgeandotheriscorrespondstoablockare
incident [8]. In [2], we generalized the concept of plick graph and were termed as generalized
abc-block edge transformation graphs Qabc(G) of a graph G and obtained 64 kinds of graphs.
In this paper, we consider one among those 64 graph which is defined as follows:
Definition 1.1 The generalized abc-block edge transformation graph Q+0 (G) is a graph whose
−
vertex set is theunion of the edges and blocks of G, in which twovertices are adjacent whenever
corresponding edges of G are adjacent or one corresponds to an edge and other to a block of G
are not incident with each other.
Generally,aSmarandachelyblock-edgeH-graphissuchagraphwithvertexsetE(G) B(G)
and two vertices e ,e E(G) B(G) are adjacent if e ,e E(H) are adjacent, or at least
1 2 ∈ 1 2 ∈ S
one of e ,e not in E(H) and they are non-adjacent, or one in E(H) and other in B(G) which
1 2 S
are not incident, where H is a subgraph of G with property P. Clearly, a Smarandachely
block-edgeE(G) B(G)-graphisnothingelsebutageneralizedabc-blockedgetransformation
graph.
S
In this paper, we study the girth, covering invariants and the domination number of
Q+0 (G). We present necessary and sufficient conditions for Q+0 (G) to be planar, outer-
− −
planar, minimally nonouterplanar and maximal outerplanar. Further, we establish a necessary
andsufficientconditionforthe generalizedabc-blockedgetransformationgraphQ+0 (G)have
−
crossing number one. Some other graph valued functions were studied in [3, 4, 7, 8, 9, 11, 12].
In Figure 1, a graph G and its generalized abc-block edge transformation graph Q+0 (G) are
−
shown.
Figure 1. Graph G and its Q+0 (G).
−
Generalizedabc-BlockEdgeTransformationGraphQabc(G)Whenabc=+0 3
−
In Q+0 (G), the vertices correspond to edges of G denoted by circles and vertices corre-
−
spondtoblocksofGdenotedbysquares. Thevertexe′ (B′)ofQ+0 (G)correspondingtoedge
i i −
e (block B ) of G and is refereed as edge (block)-vertex.
i i
The following theorems will be useful in the proof of our results.
Theorem 1.1([8]) If G is a nontrivial connected (p,q) graph whose vertices have degree d and
i
p
if b the number of blocks to which vertex v belongs in G, then P(G) has q p+1+ b
i i i
−
i=1
p X
1
vertices and d2 edges.
2 i
i=1
X
Theorem 1.2([5]) For any nontrivial connected graph G with p vertices,
α (G)+β (G)=p=α (G)+β (G).
0 0 1 1
Theorem 1.3([6]) If L(G) is the line graph of a nontrivial connected graph G with q edges,
then
q
α (L(G))= .
1
⌈2⌉
§2. Basic Results on Q+0 (G)
−
We start with preliminary remarks.
Remark 2.1 L(G) is an induced subgraph of Q+0 (G).
−
Remark 2.2 If G is a block, then Q+0 (G)=L(G) K .
− 1
∪
Remark 2.3 Let G be a graph with edge set E(G) = e ,e , ,e and r blocks. Then
1 2 m
{ ··· }
dQ+0−(G)e′i =dGei+r−1.
Remark 2.4 Let G be a (p,q)-graph with block set U(G) = B ,B , ,B such that
1 2 r
{ ··· }
|E(Bi)|=ni. Then dQ+0−(G)Bi′ =q−ni.
Theorem 2.1 Let G be a (p,q) connected graph whose vertices have degree d with r 1
i
− ≥
blocks and b (1 i p) the number of blocks to which vertex v belongs in G. Then
i i
≤ ≤
p
(1) The order of Q+0 (G) = q p+1+ b ;
− i
−
i=1
pP
(2) The size of Q+0 (G) = q(r 2)+ 1 d2.
− − 2 i
i=1
P
Proof It is shown in [5] that for a connected graph G with p vertices and b number of
i
blocks to which vertex v (1 i p) belongs in G. Then the number of blocks of G is given
i
p ≤ ≤
by b(G) = 1+ (b 1). The order of Q+0 (G) is the sum of the number of edges of G and
i −
−
i=1
P
4 K.G.Mirajkar,PoojaB.andShreekantPatil
number of blocks of G. Hence the order of Q+0 (G)
−
p p
= q+1+ (b 1)=q p+1+ b .
i i
− −
i=1 i=1
X X
The total number of edges formed by joining each of the r block-vertices to all the q
p
1
edge-vertices is rq. The number of edges in line graphL(G) is q+ d2. Thus, the size of
− 2 i
i=1
X
p p
1 1
Q+0−(G)=rq−q−q+ 2 d2i =q(r−2)+ 2 d2i. 2
i=1 i=1
X X
An immediate consequence of the above theorem is the following corollary.
Corollary 2.2 Let G be a (p,q) graph whose vertices have degree d with r blocks and m
i
components. If b (1 i p) is the number of blocks to which vertex v belongs in G, then
i i
≤ ≤
p
(1) The order of Q+0 (G) = q p+m+ b ;
− i
−
i=1
X
p
1
(2) The size of Q+0 (G) = q(r 2)+ d2.
− − 2 i
i=1
X
Theorem 2.3 Let G be a graph. The graphs Q+0 (G) and P(G) are isomorphic if and only if
−
G has two blocks.
Proof Let G be a (p,q) graph with r 1 blocks. Suppose Q+0 (G) = P(G). Then
−
≥
E(Q+0 (G)) = E(P(G)). By Theorems 1.1 and 2.1, we have
−
| | | |
p p
1 1
q(r 2)+ d2 = d2
− 2 i 2 i
i=1 i=1
X X
q(r 2) = 0.
−
Since G has at least one edge and hence equality holds only when r =2. Therefore G has
two blocks.
Conversely, suppose G has two blocks B and B . Then by definitions of Q+0 (G) and
1 2 −
P(G), L(G) is induced subgraph of Q+0 (G) and P(G). In Q+0 (G), block-vertex B is
− − 1′
adjacentalltheedge-verticescorrespondingtoedgesofB andblock-vertexB isadjacenttoall
2 2′
theedge-verticescorrespondingtoedgesofB . InP(G),block-vertexB isadjacentalltheedge-
1 1′
vertices corresponding to edges of B and block-vertex B is adjacent to all the edge-vertices
1 2′
correspondingtoedgesofB . Thisimpliesthatthereexistaone-to-onecorrespondencebetween
2
vertices of Q+0 (G) and P(G) which preserves adjacency. Therefore the graphs Q+0 (G) and
− −
P(G) are isomorphic.
2
The following theorem gives the girth of Q+0 (G).
−
Generalizedabc-BlockEdgeTransformationGraphQabc(G)Whenabc=+0 5
−
Theorem 2.4 For a graph G=2K ,K ,P ,
2 2 3
6
3 if G contains K or K or G=P ; n 4 or G is union of at least
1,3 3 n
≥
 two cycles or paths or G is union of paths and cycles,
g(Q+0−(G))= 46 iiff GG==m3KK2,,m≥4,
2
 n if G=Cn,n≥4.
Proof If G contains a triangle or K , then the line graph L(G) of G contains triangle.
1,3
By Remark 2.1, it follows that girth of Q+0 (G) is 3. Assume that G is triangle free and K
− 1,3
free. Then we have the following cases:
Case 1. Assume G has every vertex of degree is 2. We have two subcases:
Subcase1.1 IfGisconnected,thenclearlyG=C ;n 4,wehaveQ+0 (G)=C K ,
n − n 1
≥
n 4. Therefore girth of Q+0 (G) is n.
≥ − S
Subcase 1.2 If G is disconnected, then G is union of at least two cycles and Q+0 (G)
−
contains at least two wheels. Therefore girth of Q+0 (G) is 3.
−
Case 2. Assume that G=2K ,K has every vertex of degree is one. It is easy to see that
2 2
6
6 if G=3K ,
g(Q+0 (G))= 2
−

4 if G=mK ; m 4.
 2 ≥
Case 3. Assume that G = P has vertices of degree one or two. Then G is either union of
3
6
paths P or union of paths and cycles. Therefore girth of Q+0 (G) is 3.
n −
2
§3. Covering Invariants of Q+0 (G)
−
Theorem 3.1 For a connected (p,q)-graph G with r blocks, if Q+0 (G) is connected, then
−
α (Q+0 (G))=q and β (Q+0 (G))=r.
0 − 0 −
Proof Let G be a connected (p,q)-graph. By Remark 2.1, L(G) is an induced subgraphof
Q+0 (G). Therefore by definition of Q+0 (G), the edge-vertices covers all the edges of L(G).
− −
Since Q+0 (G) is connected, it follows that for each block-vertex B of Q+0 (G), there exists
− ′ −
a edge-vertexe suchthat e and B are adjacent in Q+0 (G). Therefore the vertex set of L(G)
′ ′ −
coversallthe edgesofQ+0 (G)andthis is minimumcovering. Hence α (Q+0 (G))=q. Since
− 0 −
Q+0 (G) is connected. By Theorem 1.2, we have α (Q+0 (G))+β (Q+0 (G))=q+r. Thus
− 0 − 0 −
β (Q+0 (G))=r.
0 −
2
Theorem 3.2 Let G be a connected (p,q)-graph with r blocks. If Q+0 (G) is connected, then
−
r if G is a tree,
α (Q+0 (G))=
1 −  r+⌈q−2r⌉ otherwise.
