Denise Amar, Evelyne Flandrin, Grzegorz Gancarzewicz

20 Pages·2015·0.27 MB·English

Opuscula Mathematica • Vol. 29 • No. 4 • 2009
http://dx.doi.org/10.7494/OpMath.2009.29.4.345
Denise Amar, Evelyne Flandrin, Grzegorz Gancarzewicz
CYCLABILITY IN BIPARTITE GRAPHS
Abstract. Let G = (X,Y;E) be a balanced 2-connected bipartite graph and S ⊂ V(G).
We will say that S is cyclable in G if all vertices of S belong to a common cycle in G. We
givesufficient degree conditions in a balanced bipartite graph Gand a subset S ⊂V(G) for
thecyclability of theset S.
Keywords: graphs, cycles, bipartite graphs.
Mathematics Subject Classification: 05C20, 05C35, 05C38, 05C45.
1. INTRODUCTION
We shall consider only finite graphs without loops and multiple edges.
Several authors have given results about cycles containing specific subsets of ver-
tices, see for example [7] or [9].
The setS ofverticesiscalledcyclable inGifallverticesofS belongtoacommon
cycle in G. We also speak about cyclability or noncyclability of the vertex set S.
In a bipartite graphG=(X,Y;E) we will call the independent sets of vertices X
and Y the partite sets.
Let G=(X,Y;E) be a bipartite graph and let S ⊂V(G), then S =S∩X and
X
S =S∩Y. We will say that S is balanced iff |S |=|S |.
Y X Y
In 1992 Shi Ronghua [8] obtained the following result:
Theorem 1.1. Let G be a 2-connected graph of order n and S a subset of V(G) with
|S|≥3. If for every pair of nonadjacent vertices x and y in S we have
d(x)+d(y)≥n,
then S is cyclable in G.
Note that the assumption of 2-connectivity may be omitted in Theorem 1.1. It is
an easy corollary of a result of K. Ota [7].
Recently R. Čada, E. Flandrin and Z. Ryjáček [3] proved the following generali-
zation of Theorem 1.1:
345
346 Denise Amar,EvelyneFlandrin, Grzegorz Gancarzewicz
Theorem 1.2. Let G be a 2-connected graph of order n and S a subset of V(G). If
for every pair of nonadjacent vertices x and y in S we have
d(x)+d(y)≥n−1,
then either S is cyclable in G, or n is odd and G contains an independent set S ⊆S
1
such that |S |= n and every vertex of S is adjacent to all vertices in G\S .
1 2 1 1
In2002E.Flandrin,H.Li,A.MarczykandM.Woźniak[4]obtainedthefollowing
generalizationof Theorem 1.1:
Theorem 1.3. Let G be a k-connected graph, k ≥ 2 of order n. Denote S ,...S
1 k
subsets of the vertex set V(G) and let S = S ∪S ∪···∪S . If for any x, y ∈ S ,
1 2 k i
xy 6∈E we have
d(x)+d(y)≥n,
then S is cyclable in G.
The notion of cyclability is a generalization of the term of hamiltonicity. If we
consider S = V(G) then S is cyclable iff G is hamiltonian. In fact Theorem 1.1 is a
generalizationof the following result of O. Ore [6]:
Theorem 1.4. Let G be a graph on n ≥ 3 vertices. If for all nonadjacent vertices
x,y ∈V(G) we have
d(x)+d(y)≥n,
then G is hamiltonian.
A similar result for bipartite graphs was proved by J. Moon and M. Moser [5]
in 1963:
Theorem 1.5. Let G=(X,Y;E) be a balanced bipartite graph of order 2n. If for all
nonadjacent vertices x∈X and y ∈Y we have
d(x)+d(y)≥n+1,
then G is hamiltonian.
Givenabalancedbipartitegraphandaselectedsubsetofvertices,weareinterested
in properties that imply cyclability.
In 2000 D. Amar, M. El Kadi Abderrezzak, E. Flandrin [2] proved the following
generalizationof Theorem 1.1 for bipartite graphs:
Theorem 1.6. Let G=(X,Y;E) be a balanced 2-connected bipartite graph of order
2n, S ⊂X. If for every x∈S, y ∈Y, xy 6∈E we have
d(x)+d(y)≥n+1,
then S is cyclable in G.
Note that in this case S = S and Theorem 1.6 is also a generalization of
X
Theorem 1.5.
The main result of the present paper, given in Section 3, is Theorem 3.1, which
improves upon Theorem 1.6.
Cyclability in bipartite graphs 347
2. DEFINITIONS
Let G be a graph and H a subgraph of G.
Definition 2.1. NG(H) denotes the set of all vertices of the graph G which are
adjacent to a vertex of the subgraph H, i.e. NG(H)={u∈V(G) : ∃v ∈V(H) such
that uv ∈E(G)}.
Consider an arbitrary vertex x ∈ V(G). N(x) denotes the set of all neighbors of
the vertexxinG, i.e. N(x)={u∈V(G): xu∈E(G)}.NH(x) denotesthe setofall
neighborsofthevertexxinthesubgraphH,i.e. NH(x)={u∈V(H): xu∈E(G)}.
d (x) denotes the number of neighbors of x in the subgraph H i.e.
H
d (x)=|N (x)|, and d (x) denotes the degree of the vertex x in the subgraph H.
H H H
Intheproofwewillonlyusecyclesandpathswithagivenorientation. Foracycle
C :c ...c or a path P :p ...p we will use implicit orientation.
1 k 1 l
Thus it makes sense to speak of a successor ci+1 and a predecessor ci−1 of a
vertex c (addition modulo l+1). Denote the successor of a vertex x by x+ and its
i
predecessor by x−. This notation can be extended to A+ = {x+ : x ∈ A}, and
similarly, to A− when A⊆V(G).
Let P be a path p ...p and u, v ∈V(G) such that up , vp ∈E(G), then: uPv
1 k 1 k
is the path up ...p v and vPu is the path vp ...p u.
1 k k 1
Definition 2.2. We shall call a path P :p ...p a C-path of the cycle C iff V(P)∩
1 l
V(C)={p ,p }. Note that a C-path is a generalized chord of the cycle.
1 l
Definition 2.3. Let C : c ...c be a cycle in G with the orientation, the indices
1 l
1,...,l are considered modulo l. For any pair of vertices c , c ∈ V(C) (i 6= j) we
i j
define four intervals:
– ]ci,cj[ is the path ci+1...cj−1.
– [ci,cj[ is the path ci...cj−1.
– ]c ,c ] is the path c ...c .
i j i+1 j
– [c ,c ] is the path c ...c .
i j i j
Note that these four intervals are subsets of the cycle C.
For notation and terminology not defined above a good reference is [1].
3. THEOREM
Theorem 3.1. If G = (X,Y;E) is a balanced 2-connected bipartite graph of order
2n and S ⊂V(G) satisfying conditions:
For every x∈S , y ∈Y, xy 6∈E we have d(x)+d(y)≥n+1 (3.1)
X
For every x∈X, y ∈S , xy 6∈E we have d(x)+d(y)≥n+1 (3.2)
Y
then S is cyclable in G.
348 Denise Amar,EvelyneFlandrin, Grzegorz Gancarzewicz
K
2,p
K
2,2 K
2,p
K
3p,6
K
2,p
S vertices from X vertices from Y
Fig. 1. Gp,3
Theorem 3.1 is obviously a generalizationof Theorem 1.6.
We first tried to find a generalizationsatisfying two conditions:
– The vertices of S are in both partite sets X and Y.
– The degree sum condition holds only for vertices from S.
However, even if we assume that S is balanced (i.e. |S |=|S |), such a result is
X Y
not true.
Foreveryk ≥1wewillgiveanexampleofa2-connected,balancedbipartitegraph
G=(X,Y;E) and a balanced set S ⊂V(G), satisfying the following condition:
For every x∈S , y ∈S if xy 6∈E then d(x)+d(y)≥n+k, (3.3)
X Y
such that S is not cyclable in G.
p−k
Let k ≥1, p≥k+2 and 2≤r ≤1+ .
2
Cyclability in bipartite graphs 349
First consider bipartite graphs K , K and r copies of K . In K we have
pr,2r 2,2 2,p 2,2
two partite sets say X and Y . The graph G = (X,Y;E) is obtained out of K ,
2 2 pr,2r
K and the r copies of K by joining every vertex of degree pr from K with
2,2 2,p pr,2r
all vertices from X and every vertex of degree p from the r copies of K with all
2 2,p
vertices from Y .
2
Let S be the set of all vertices from K of degree pr in K .
1 pr,2r pr,2r
In each copy of K we take the two vertices of degree p in K . In this way we
2,p 2,p
will get 2r vertices and we define the set S as the set containing these 2r vertices.
2
We can define now the set S. Let S =S ∪S ∪V(K ).
1 2 2,2
p−k
For k ≥ 1, p ≥ k + 2 and 2 ≤ r ≤ 1+ we have obtained a balanced,
2
2-connected bipartite graph G =(X,Y;E) of order 2n with n=pr+2r+2 and a
p,r
balanced set S which is not cyclable, but satisfies (3.3).
We can find an example of the graph G on the Figure 1.
p,3
This example shows that it is not enough to assume that the degree sum con-
dition holds only for the vertices from S in a bipartite graph. Even increasing the
connectivity will not be sufficient, as we can see in the following example.
For every k ≥ 1 and l ≥ 2 we will give an example of an l-connected, balanced
bipartite graph G′ = (X,Y;E) and a balanced set S′ ⊂V(G′), satisfying (3.3), such
that S′ is not cyclable in G′.
p−k
Let k ≥1, l≥2, p≥l2−l+k and l≤r <1+ .
l
First consider bipartite graphs K , K and r copies of K . In K we have
pr,lr l,l l,p l,l
two partite sets say X and Y . The graph G′ = (X,Y;E) is obtained out of K ,
l l pr,lr
K and the r copies of K by joining every vertex of degree pr from K with
l,l l,p pr,lr
all vertices from X and every vertex of degree p from the r copies of K with all
l l,p
vertices from Y.
l
Let S′ be the set of all vertices from K of degree pr in K .
1 pr,lr pr,lr
In eachcopy of K we take the l vertices of degree p in K . In this way we will
l,p l,p
get lr vertices and we define the set S′ as the set containing these lr vertices.
2
We can define now the set S′. Let S′ =S′ ∪S′ ∪V(K ).
1 2 l,l
p−k
Fork ≥1,l≥2,p≥l2−l+kandl≤r <1+ wehaveobtainedabalanced,
l
l-connected bipartite graph G′ =(X,Y;E) of order 2n with n=pr+lr+l and a
p,r,l
balanced set S′ which is not cyclable in G′, but satisfies (3.3).
4. PROOF OF THEOREM 3.1
4.1. PRELIMINARY NOTATIONS
Let G=(X,Y;E) be a bipartite graph and let C be a cycle in G.
InthischapterforagivencycleC andavertexx∈V(G\C),aC-pathQthrough
xwillbedenotedQ:uQ xQ u′,whereQ andQ aretwovertexdisjointpaths. The
1 2 1 2
endverticesofthe C-pathQ:u andu′ andthe vertexx donotbelong to Q norQ .
1 2
350 Denise Amar,EvelyneFlandrin, Grzegorz Gancarzewicz
Note that the path Q may be empty or in other words V(Q ) = ∅ and in this
1 1
case xu∈E. Similarly for Q .
2
AnexampleofaC-pathP :uP xP u′throughavertexxcanbefoundonFigure2.
1 2
x
P P
1 2
u
u 1 u′
u
2
C
Fig. 2. An example of a cycle C and a C-path P with x,u′, u1 ∈X and u, u2 ∈Y
Remark 4.1. Given a 2-connectedgraph G, a nonhamiltoniancycle C and a vertex
x∈V(G\C), G contains necessarily a C-path through x.
In the remainingpartofSection4 we willalwaysconsidera 2-connectedbipartite
graph G and a subset S ⊂ V(G) not cyclable in G. Given a cycle C, a vertex
x∈V(G\C)∩S suchthatC containsS\{x}butdoesnotcontainS,wewilldenote
by P a C-paththroughx. We willalwaysassumethat the cycleC andthe C-pathP
arechosensuchthatP isshortestpossibleamongallC-paths throughxforallcycles
C containing S \{x}, i.e. for any cycle C′ containing S \{x}, and for any C′-path
P′ containing the vertex x we have |V(P)| ≤ |V(P′)|. We will denote this C-path
P :uP xP u′ (note that P and/or P may be empty).
1 2 1 2
Wewilldenotebyu thefirstvertexonthecycleC fromS afteru(u existssince
1 1
S is not cyclable). Similarly u is the first vertex on the cycle C from S after u′.
2
R is the subgraph induced in G by V(G)\V(C).
Alltheintervalsoftype[a,b],[a,b[,]a,b]and]a,b[areintervalsonthecycleC and
we sometimes identify the vertex set of an interval with the corresponding interval.
Remark 4.2. The C-path P :uP xP u′ has the following properties:
1 2
– If V(P )6=∅ or V(P )6=∅ then d (x)≤1. (4.1)
1 2 C
– If V(P )6=∅ and V(P )6=∅ then d (x)=0. (4.2)
1 2 C
Remark 4.2 is an immediate consequence of the choice of the cycle C and the
C-path P.
Cyclability in bipartite graphs 351
4.2. FORMULATION AND PROOF OF LEMMA 4.3
In the proofof Theorem3.1 we shalluse the following lemma. Notations G, S, C, R,
x and C-path P :uP xP u′, u and u are defined in Section 4.1 but we recall them
1 2 1 2
for completeness. We denote by C a cycle containing S \{x}. Let P be a C-path
throughx. ThecycleC andthe C-pathP arechosensuchthatP is shortestpossible
among all C-paths through x for all cycles C containing S\{x}. Let u be the first
1
vertexfromS after u onthe cycleC, andletu be the firstvertexfromS after u′ on
2
the cycle C. The subgraph of G induced by V(G)\V(C), will be denoted R.
Lemma 4.3. Let G=(X,Y;E)be a 2-connectedbipartite graph and let C, P, R and
S be as above. Then we have:
′
– For every C−path Q:aQ xQ a through x we have:
1 2
′ ′
V(]a,a[)∩S 6=∅ and V(]a,a[)∩S 6=∅. (4.3)
′
– For any b∈V(]u,u ]) and c∈V(]u,u ]) we have:
1 2
NP xP (b)=NP xP (c)=∅. (4.4)
1 2 1 2
′
– If z ∈V(P)\{u,u}, then NR(z)∩(N(]u,u1])∪N(]u,u2]))=∅. (4.5)
′
– NR(]u,u1])∩NR(]u,u2])=∅. (4.6)
– For any y ∈NR(x) we have N]u,u ](y)=N]u′,u ](y)=∅. (4.7)
1 2
Proof of Lemma 4.3. Suppose that V(]a,a′[)∩S =∅, then the cycle:
C′ : aQ xQ a′a′+... a, (4.8)
1 2
is a cycle containing S, a contradiction. If V(]a′,a[) ∩ S = ∅, then using similar
arguments we get a contradiction and hence (4.3) is proved.
In order to prove (4.4) suppose that there is a vertex b ∈ V(]u,u ]) such that
1
NP xP (b) 6= ∅. We have a vertex z ∈ NP xP (b) and we assume that the vertices
1 2 1 2
on the path P xP are labeled as follows: P xP : p1...plxpk...p1.
1 2 1 2 1 1 2 2
We shall consider three cases.
1. When z =x, then the following cycle:
C′ : uP xbb+...u′...u ...u, (4.9)
1 2
contains S, a contradiction.
2. If z ∈V(P ), then the following cycle:
1
C′ : up1...pizbb+...u ...u′...u ...u, (4.10)
1 1 1 2
contains S\{x} and has a C′-path
P′ : zpi+2...xP u′, (4.11)
1 2
shorter then P, a contradiction with the choice of C and P.
352 Denise Amar,EvelyneFlandrin, Grzegorz Gancarzewicz
3. If z ∈V(P ), then the following cycle:
2
C′ : uP xpk...pjzbb+...u′...u ...u, (4.12)
1 2 2 2
contains S, a contradiction.
So we have NP xP (b) = ∅. Using similar arguments we can prove that for any
1 2
c∈V(]u′,u2]) NP xP (c)=∅, and hence (4.4) is true.
1 2
We will prove now (4.5).
Suppose that NR(z)∩(N(]u,u1])∪N(]u′,u2]))6=∅.
So we have a vertex w ∈ NR(z) ∩ (N(]u,u1]) ∪ N(]u′,u2])). Without loss of
generality we can assume that w ∈NR(z)∩N(]u,u1]). Let a ∈V(]u,u1]), such that
aw∈E. From (4.4) we know that w6∈V(P xP ).
1 2
x
P P
1 z 2
w
u
u 1 u′
u
2
C
Fig. 3. u2, w, x∈X and u, u1, u′, z∈Y
As in the proof of (4.4), we shall consider three cases:
1. z ∈V(P ).
1
2. z ∈V(P ).
2
3. z =x.
Using similar arguments we get contradiction.
In any case we obtain a contradictionby replacing in (4.10), (4.12)and (4.9), the
edge za by the path zwua. Hence (4.5) is true.
For a = u , you can find the illustrations of Cases 1 and 2 on Figures 3 and 4
1
respectively.
In order to prove (4.6), suppose that b ∈ V(]u,u ]), c ∈ V(]u′,u ]) and z ∈
1 2
NR({b,c}).
From (4.4) we know that N (b)=N (P xP )=∅, and so z 6∈V(P). In this
P xP c 1 2
1 2
case the cycle:
C′ : uP xP u′u′−...u ...bzc...u u+...u
1 2 1 2 2
contains S, a contradiction.
Cyclability in bipartite graphs 353
x
z
P P
1 2
w
u
u 1 u′
u
2
C
Fig. 4. u2, w, x∈X and u, u1, u′, z∈Y
Hence (4.6) is true.
In order to prove (4.7), suppose that there is a vertex y ∈ NR(x) such that
N (y) 6= ∅. From (4.4) we know that y 6∈ V(P xP ). We have a vertex b ∈
]u,u ] 1 2
1
V(]u,u ]) such that yb∈E and the following cycle:
1
C′ : uP xybb+...u ...u′...u ...u
1 1 2
contains S, a contradiction and so N (y)=∅.
]u,u ]
1
Using the same arguments we can prove N (y)= ∅. Hence (4.7) is true and
]u,u ]
2
the proof of Lemma 4.3 is finished.
(cid:3)
4.3. PROOF OF THEOREM 3.1
We mayassumethatS 6=∅and|S |≥|S |. We willproceedbyinductionoverthe
Y Y X
number of vertices in S .
X
If |S |=0, then S =S and from Theorem1.6 we know that S is cyclable in G.
X Y
So the first step of the induction is finished.
Suppose now that S satisfies the assumptions of Theorem 3.1 and |S |≥1.
X
From the induction hypothesis, we assume that for any x∈S the set S\{x} is
X
cyclable in G, while S itself is not cyclable. Let us choose a vertex x∈S .
X
WehaveacycleC containingS\{x}suchthatx6∈V(C). Werecallthatthecycle
C and the C-path P are chosen such that P is shortest possible among all C-paths
containing x for all cycles C containing S \{x}. As in Section 4.1, u is the first
1
vertex from S on the cycle C after u and u is the first vertex from S on the cycle C
2
after u′, R is the subgraph induced in G by V(G)\V(C).
It is clear that in this case R is a balanced bipartite graph.
c+r
Note that if c=|V(C)|, r=|V(R)|, then c and r are even and n= .
2
From Remark 4.2 and Lemma 4.3 C and P satisfy (4.3) — (4.7).
354 Denise Amar,EvelyneFlandrin, Grzegorz Gancarzewicz
We shall consider four cases:
1. NR(x)=∅.
2. NR(x)6=∅ and u1,u2 ∈SY.
3. NR(x)6=∅ and u1, u2 ∈SX.
4. NR(x)6=∅ and u1 and u2 are in different partite sets.
Case 1. NR(x)=∅
In this case P and P are empty and xu, xu′ ∈E.
1 2
Since R is balanced there is an y ∈ Y ∩V(R). Since xy 6∈ E then from (3.1) we
have:
d(x)+d(y)≥n+1. (4.13)
Since NR(x)=∅ we have:
r
d (x)=0 and d (y)≤ −1. (4.14)
R R
2
x y
a a+ b b+
C
Fig. 5. a+,b+, x∈X and a,b, y∈Y
Suppose that y has two neighbors a+, b+ in NC(x)+, then xa, xb ∈ E and the
cycle C′ (see Fig. 5):
C′ : xbb−...a+yb+b++...ax
contains S, a contradiction with noncyclability of S. So y has at most one neighbor
in N (x)+ and thus:
C
c
d (x)+d (y)≤ +1. (4.15)
C C
2
From (4.14) and (4.15) we have:
r c
d(x)+d(y)≤ −1+ +1≤n,
2 2
a contradiction with (3.1).