An Extension of Cui-Kano's Characterization Problem on Graph Factors

0.13 MB·English

by  Hongliang Lu

An Extension of Cui-Kano’s Characterization Problem on
3 Graph Factors
1
0
2
Hongliang Lu∗
n
a
School of Mathematics and Statistics
J

6 Xi’an Jiaotong University, Xi’an 710049,PR China
2

]
O
C
Abstract
.
h
t Let G be a graph with vertex set V(G) and let H : V(G) → 2N be a set function
a
m associating with G. An H-factor of graph G is a spanning subgraphs F such that
[
dF(v)∈H(v) for every v ∈V(G).
2
v
Let f :V(G)→N be an eveninteger-valuedfunction such that f ≥4 and let H (v)=
f
7
5 {1,3,...,f(v) − 1,f(v)} for v ∈ V(G). In this paper, we investigate Hf-factors of
6 graphs G by using Lov´asz’s structural descriptions. Let o(G) denote the number of
4
odd components of G. We show that if one of the following conditions holds, then G
.
1
contains an H -factor.
0 f
3
(i) o(G−S)≤f(S) for all S ⊆V(G);
1
v: (ii) |V(G)| is odd, dG(v) ≥ f(v)−1 for all v ∈ V(G) and o(G−S) ≤ f(S) for all
i ∅6=S ⊆V(G).
X
r As a corollary, we show that if a graph G with odd order and minimum degree 2n−1
a
satisfies
o(G−S)≤2n|S| for all ∅6=S ⊆V(G),
then G contains an H -factor. In particular, we make progress on the characterization
n
problem for a special family of graphs proposed by Akiyama and Kano.
1 Introduction
All graphs in this paper are simple. Let G be a graph with vertex set V(G) and edge set
E(G). We denote the degree of v in G by d (v). The minimum degree in graph G will be
G
denoted by δ(G) and the maximum degree by △(G). The subgraph induced by the set S
∗
Corresponding email: [email protected] (H.Lu)
1
is denoted by G[S]. The number of components of graph G is denoted by ω(G) and the
number of odd components of G by o(G). Let E (S,T) denote the set of edges of graph G
G
withoneendinS andtheother endinT ande (S,T) =|E (S,T)|. Thejoin G= G +G ,
G G 1 2
is the graph obtained from two vertex disjoint graphs G and G by joining each vertex in
1 2
G to every vertex in G .
1 2
Let H be a function associating a subset of Z to each vertex of G. A spanning subgraph
F of graph G is called an H-factor of G if
d (x) ∈ H(x) for every vertex x ∈ V(G). (1)
F
By specifying H(x) to be an interval or a special set, an H-factor becomes an f-factor, an
[a,b]-factor or a (g,f)-factor, respectively.
Let F be a spanning subgraph of G. Following Lov´asz [8], one may measure the “devi-
ation” of F from the condition (1) by
∇H(F) = X min(cid:8)|dF(v)−h| : h∈ H(v)(cid:9). (2)
v∈V(G)
Moreover, the “solvability” of (1) can be characterized by
∇(H) = min{∇ (F) : F is a spanning subgraph of G}.
H
ThesubgraphF is said tobeH-optimal if ∇ (F) = ∇(H). Itis clear that F is an H-factor
H
if and only if ∇ (F) = 0, and any H-factor (if exists) is H-optimal. Let
H
Q = {h ,h ,...,h },
1 2 m
where h < h < ··· < h . Then Q is called an allowed set if each of the gaps of Q has at
1 2 m
most one integer, i.e.,
h −h ≤ 2 for all 1 ≤ i≤ m−1.
i+1 i
A set function H associating with G is called an allowed set function (following [8]) if H(v)
is an allowed set for all v ∈ V(G).
Lov´asz [8]showed thatif H is notan allowed set, then thedecision problemof determin-
ing whether a graph has an H-factor is known to be NP-complete. Cornu´ejols [3] provided
the first polynomial algorithm for the problem with H allowed.
A special case of H-factor problem is the so-called (1,h)-odd factor problem, i.e., the
problem with
H(v) = {1, 3,..., h(v)−2, h(v)},
2
where h: V(G) → N be an odd function. For a constant odd integer n≥ 1, if h(x) = n for
all x ∈ V(G), then (1,h)-odd factor is called (1,n)-odd factor. The first investigation of the
(1,n)-odd factor problem is due to Amahashi [2], who gave a Tutte type characterization
for graphs having a global odd factor.
Theorem 1.1 (Amahashi) Let n be an odd integer. A graph G has an (1,n)-odd factor
if and only if
o(G−S) ≤ n|S| for all subsets S ⊂ V(G). (3)
For general odd value functions h, Cui and Kano [4] established a Tutte type theorem.
Theorem 1.2 (Cui and Kano, [4]) Let h : V(G) → N be odd value function. A graph
G has an (1,h)-odd factor if and only if
o(G−S) ≤ h(S) for all subsets S ⊂ V(G). (4)
Noticingtheformofthecondition(4),theyaskedthequestionofcharacterizinggraphsG
in terms of graph factors such that
o(G−S)≤ 2n|S| for all subsets S ⊂ V(G). (5)
MotivatedbyCui-Kano’sproblem,LuandWang[9]considerthedegreeprescribedsubgraph
problem for the special prescription
H = {1,3,...,2n−1,2n}. (6)
n
Theorem 1.3 (Lu and Wang, [9]) Let G be a connected graph. If
o(G−S)≤ 2n|S| for all subsets S ⊂ V(G), (7)
then G contains an H -factor.
n
The condition of Theorem 1.3 implies that |V(G)| is even. Let H∗ = H ∪{−1}. For odd
n n
ordergraph,theyobtainedthefollowingresult(forconvenience, thedefinitionofH∗-critical
n
graph will be introduced in Section 2).
Theorem 1.4 (Lu and Wang, [9]) Let G be a connected graph of odd order. Suppose
that
o(G−S) ≤ 2n|S| for all ∅ =6 S ⊂ V(G). (8)
Then either G contains an H -factor, or G is H∗-critical.
n n
3
The condition (4) implies that the graph is even order. For odd order graph, Akiyama
and Kano propose the following problem (see also [1, Problem (6.14)] ).
Problem 1.5 (Akiyama and Kano, [1]) Let G be a connected graph and h :V(G) → N
be an even integer-valued function. If G satisfies
o(G−S) ≤ h(S) for all ∅ =6 S ⊂ V(G), (9)
what factor or property does G has?
Let f ≥4 be an even integer-value function and let H : V(G) → 2N be an set function
f
such that H (v) = {1,3,...,f(v)−1,f(v)} for v ∈ V(G). Motivated by Akiyama-Kano’s
f
problem, we investigate the structure of graphs without H -factor by using Lov´asz’s H-
f
factorstructuretheory[8]. Weobtainthefollowingresult,whichisanextensionofTheorem
1.3.
Theorem 1.6 Let G be a graph with even order. If
o(G−S)≤ f(S) for all S ⊂ V(G), (10)
then G contains an H -factor.
f
The inequality (10) also implies that |V(G)| is even. For odd order graph, we solve
Problem 1.5 and obtain a stronger result than Theorem 1.4.
Theorem 1.7 Let G be a connected graph with odd order. Suppose that d (v) ≥ f(v)−1
G
for all v ∈ V(G). If
o(G−S) ≤ f(S) for all ∅ =6 S ⊂ V(G), (11)
then G contains an H -factor.
f
Corollary 1.8 Let n ≥ 2 be an integer and let G be a connected graph with odd order and
minimum degree 2n−1. If
o(G−S) ≤ 2n|S| for all ∅ =6 S ⊂ V(G), (12)
then G contains an H -factor.
n
Remark 1: In Corollary 1.8, the conditions “δ(G) ≥ 2n−1” is sharp. Let K denote
2n−1
the complete graph of order 2n − 1. Take 2n − 2 disjoint copies of K . Add a new
2n−1
4
vertices v and connect two vertices in each copy of K to the new vertex v. This results
2n−1
a connected graph G with odd order (2n−2)(2n−1)+1 and minimum degree 2n−2. It
is easy to show that
o(G−S) ≤ 2n|S| for all ∅ =6 S ⊂ V(G). (13)
Now we show that G contains no H -factor. Otherwise, suppose that G contains an H -
n n
factor F. By parity, K contains no H -factors and so F contains exactly an edge from
2n−1 n
a copy of K . Then we have d (v) = 2n−1 ∈/ H , a contradiction.
2n−1 F n
Remark 2: In Corollary 1.8, the condition (12) is not necessary for the existence of an
H -factor in a graph. Let m ≥ 2n+2 be an even integer. Consider the graph
n
G = K +mK
1 2n+1
obtained by linking a vertex v to all vertices in 2n+1 copies of the complete graph K .
2n+1
Clearly, G is a graph with odd order and minimum degree 2n+1. It is easy to verify that
G contains an H -factor. However, taking the subset S to be the single vertex v, we see
n
that the condition (12) does not hold for G.
2 On H-critical Graphs
In this section, we study H-factors of graphs based on Lov´asz’s structural description to
the degree prescribed subgraph problem. Denote by I (v) the set of vertex degrees in all
H
H-optimal subgraphs of graph G, i.e.,
I (v) = {d (v) : all H-optimal subgraphs F}.
H F
Comparing the set I (v) with H, one may partition the vertex set V(G) into four classes:
H
C = {v ∈ V(G) : I (v) ⊆ H(v)},
H H
A = {v ∈ V(G)−C : minI (v) ≥ maxH(v)},
H H H
B = {v ∈ V(G)−C : maxI (v) ≤ minH(v)},
H H H
D = V(G)−A −B −C .
H H H H
It is clear that the 4-tuple (A ,B ,C ,D ) is a pairwise disjoint partition of V(G). We
H H H H
call it the H-decomposition of G. In fact, the four subsets can bedistinguished according to
the contributions of their members to the deviation (2). A graph G is said to be H-critical
if it is connected and D = V(G). For non-consecutive allowed set function, the only
H
5
necessary condition of H-critical graph is given by Lov´asz [8]. In this paper, we obtain a
sufficient condition for H-critical graph.
We write MH(x) = maxH(x) and mH(x) = minH(x) for x ∈ V(G). For S ⊆
V(G), let MH(S) = MH(x) and mH(S) = mH(x). By the definition of
Px∈S Px∈S
A ,B ,C ,D , the following holds:
H H H H
(I) for every x ∈ B , there exists an H-optimal graph F such that d (x) < mH(x);
H F
(II) for every x ∈ A , there exists an H-optimal graph F such that d (x) > MH(x);
H F
(III) for every x ∈ D , there exists an H-optimal graph F such that d (x) < MH(x) and
H F
other H-optimal graph F′ such that d (x) > mH(x).
F
Lov´asz [8] gave the following properties.
Lemma 2.1 (Lov´asz, [8]) IfGisasimple graph, thenI (v)isanintervalforallv ∈ D .
H H
Lemma 2.2 (Lov´asz, [8]) The intersection I (v)∩H(v) contains no consecutive integers
H
for any vertex v ∈ D .
H
Given an integer set P and an integer a, we write P −a = {p−a | p ∈ P}. Let C be a
connected induced subgraph of G and T ⊆ V(G)−V(C). Let H : V(C) → 2N be a set
C,T
function such that H (x) = H(x)−e (x,T) for all x ∈ V(C).
C,T G
Lemma 2.3 (Lov´asz, [8]) Every component R of G[D ] is H -critical and if F is
H R,BH
H-optimal, then F[V(R)] is H -optimal.
R,BH
Lemma 2.4 (Lov´asz, [8]) If G is H-critical, then ∇(H) = 1.
Lemma 2.5 (Lov´asz, [8]) For any H-optimal graph F, E (B ,B ∪C ) ⊆ E(F), and
G H H H
E (A ,C ∪A )∩E(F) = ∅.
G H H H
Theorem 2.6 (Lov´asz, [8]) ∇(H) = ω(G[D ])+ (mH(v)−d (v))− MH(v).
H Pv∈BH G−AH Pv∈AH
In the proof of main theorems, we need the following two technical lemmas.
Lemma 2.7 Let F be an H-optimal subgraph. For every component R of G[D ], F misses
H
at most an edge of E (V(R),B ).
G H
6
Proof. Let F be an H-optimal subgraph of G. We write τ = ω(G[D ]) and G[D ] =
H H H
C ∪ ··· ∪ C . Since C is H -critical, then C contains no H -factors. So if
1 τH i Ci,BH i Ci,BH
def (C ) = 0, then F either misses at least an edge of E(C ,B ) or contains at least an
F i i H
edges of E(C ,A ). Let τ denote the number of components of G[D ] such that F misses
i H B H
at least an edge of E(C ,B ) and τ denote the number of the components of G[D] such
i H A
that F contains at least an edge of E(C ,A ). Let τ denote the number of components C
i H c i
of G[D ] such that F contains at least one edge of E(C ,A ) and misses at least one edge
H i H
E(C ,B ). Then we have
i H
∇H(F) ≥ τH −τA−τB +τc+ X min{|r−dF(x)| | r ∈ H(x)}
x∈AH∪BH
≥ τH −τA−τB +τc+ X (dF(x)−MH(x))+ X (mH(x)−dF(x))
x∈AH x∈BH
≥ τH −τA−τB +τc+(eF(AH,BH)+τA−MH(AH))+ X (mH(x)−dF(x))
x∈BH
= τH −τB +τc+(eF(AH,BH)−MH(BH))+ X (mH(x)−dF(x))
x∈BH
≥ τH −τB +τc+(eF(AH,BH)−MH(AH))+(mH(BH)−(eF(AH,BH)+ X dG−AH(x)−τB))
x∈BH
= τH(AH,BH)+τc+mH(BH)−MH(AH)− X dG−AH(x) ≥ ∇(H).
x∈BH
Since ∇ (F) = ∇(H), then we obtain τ = 0 and
H c
X dF(x) = eF(AH,BH)+ X dG−AH(x)−τBH,
x∈BH x∈BH
which implies that F misses at most an edge from C to B. This completes the proof for
i
1 ≤ i≤ τ . 2
H
Lemma 2.8 Let G be a graph and let H : V(G) be an allowed set function. If MH(v)−1 ∈
H(v) and d (v) ≥ MH(v)−1 for all v ∈ V(G), then G is not H-critical.
G
Proof. By contradiction, we firstly assume that G is H-critical. Let F be an H-optimal
subgraph of G such that E(F) is maximal.
SinceGisH-criticalandF isH-optimal,thenbyLemma2.4,wehaved (v) ≤ MH(v)+
F
1 for all v ∈ V(G). We claim that there exists a vertex x ∈ V(G) such that d (x) =
F
MH(x)+1. Otherwise, suppose that d (v) ≤ MH(v) for all v ∈ V(G). Then there exists
F
a vertex v ∈ V(G) such that d (v) ∈/ H(v) and so 0 ≤ d (v) ≤ f(v) − 2. Hence there
F F
exists an edge e ∈ E(G) − E(F), which is incident with vertex v. Then F ∪ {e} is also
H-optimal, contradicting to themaximality of F. Thus there exists a vertex x ∈V(G) such
7
that d (x) = MH(x)+1. Since I (x) is an interval and I (x)∩H(x) contains no two
F H H
consecutive integers, then we have minI (x)≥ MH(x), contradicting to x ∈ D .
H H
This completes the proof. 2
Corollary 2.9 Let n ≥ 2 be an integer and let G be a graph. If δ(G) ≥ 2n−1, then G is
not H -critical.
n
3 The Proof of Theorems 1.6 and 1.7
In this section, we assume that f : V(G) → Z+ be an even integer-valued function such
that f ≥4 and H (v) = {1,3,...,f(v)−1,f(v)} for all v ∈ V(G).
f
Theorem 3.1 Let G be a graph and let A , B , C and D be defined as above. Then
H H H H
f f f f
(a) E (B ,B ∪C )= ∅;
G H H H
f f f
(b) For every component R of G[D ], |V(R)|+|E (V(R),B )| ≡ 1 (mod 2);
H G H
f f
(c) every component R of G[D ∪B ] is odd.
H H
f f
Proof. Firstly, we prove (a) by contradiction. Suppose that there exists an edge e ∈
E (B ,B ∪C ). Without loss of generality, we assume that e = uv and u ∈ B . For
G H H H H
f f f f
any H-optimal graph F, by Lemma 2.5, e ∈ E(F) and so d (v) ≥ 1, contradicting to the
F
definition of B . This completes the proof of (a).
H
f
Secondly, we prove (b). By Lemma 2.3, R is H -critical. For simplicity, we write
R,BHf
H = H . We claim that f(u)−e (u,B ) ∈/ I (u) for all u ∈ V(R). Otherwise,
R R,BHf G Hf HR
suppose that there exists a vertex x ∈ V(R) such that f(x)−e (x,B ) ∈ I (x). By
G Hf HR
Lemma2.2,I (x)∩H (x)doesnotcontain twoconsecutive integersandsowehavef(x)−
HR R
1−e (u,B ) ∈/ I (x). ByLemma2.1,I (x)isanintervalandsowehaveminI (x) ≥
G Hf HR HR HR
MH (x), contradicting to the definition of H-critical graphs. Hence I (u) ⊆ [0,f(u)−
R HR
1−e (u,B )]. Let F be an H -optimal graph and F∗ = F[V(R)]. By Lemma 2.3, F∗
G Hf f
is an H -optimal subgraph of graph R. Furthermore, by Lemma 2.3, R is H -critical and
R R
so there exists a vertex x ∈ V(R) such that dF∗(x) ∈/ HR(x) and dF∗(y) ∈ HR(y) for all
y ∈ V(R)−x.
8
Hence for every vertex y ∈ V(R) − x, dF∗(y) ≡ f(y) − 1 − eG(y,B) (mod 2) and
dF∗(x) ≡ f(x)−2−eG(x,BH ) (mod 2). Then
f
X dF∗(v) ≡ X (f(y)−1−eG(y,BH ))+f(x)−2−eG(x,BH ) (mod 2)
f f
v∈V(R) y∈V(R)−x
≡ X eG(y,BH )+|V(R)|−1,
f
y∈V(R)
which implies
X eG(y,BH )+|V(R)| ≡ 1 (mod 2).
f
y∈V(R)
This completes the proof of (b).
Finally, we prove (c). We write B = {v ,...,v } and G[D ] = C ∪···∪C . For
Hf 1 |BHf| Hf 1 τ
1 ≤ i ≤ |B | and 1 ≤ j ≤ τ, we claim e (v ,V(C )) ≤ 1. Otherwise, suppose that there
H G i j
f
exists v ∈ B and a component C of G[D ] such that e (v,V(C )) ≥ 2. For arbitrary
H i H G i
f f
H -optimal graph F, by Lemma 2.7, then we have d (v) ≥ 1, contradicting v ∈ B .
f F Hf
Let R be an arbitrary connected component of G[D ]. Without loss of generality, we
H
f
write V(R) = C ∪···∪C ∪B , where B = {y ,...,y }. Now we construct a graph R∗
1 k 1 1 1 r
obtained from R by contracting C to a vertex x for 1 ≤ i ≤ k. By (a), R∗ is a bipartite
i i
graph.
Claim 1. R∗ is a tree.
Since R is connected, then R∗ is connected. Now we show that R∗ contains no cycles.
Conversely, suppose that R∗ contains a cycle x y ,...,x y x . We write W = C ∪···∪
1 1 m m 1 i1
C and B = {y ,...,y }. By Lemma 2.7, for any H-optimal graph F, F contains at
im 2 1 m
least m edges from W to B . Now we claim that d (v) = 1 for all v ∈B , otherwise, there
2 F 2
exists a vertex v ∈ B such that d (v) ≥ 2 contradicting to v ∈ B ⊆ B . Since F is
2 F 2 Hf
an arbitrary H -optimal graph and d (v) = 1 for all v ∈ B , then we have B ⊆ C , a
f F 2 2 Hf
contradiction again. This completes Claim 1.
LetW∗ =V(C )∪···∪V(C ). ByClaim1,R∗isatree,whichimpliesthate (W∗,B )=
1 k G 1
k+r−1. By (b), we have
k
k ≡ X(|V(Ci)|+eG(V(Ci),BH )) (mod 2)
f
i=1
k
= X|V(Ci)|+eG(W∗,B1)
i=1
k
= X|V(Ci)|+k+r−1,
i=1
9
which implies
k
X|V(Ci)|+r ≡ 1 (mod 2).
i=1
Hence |V(R)| is odd. This completes the proof. 2
By Theorems 2.6 and 3.1, we obtain the following result.
Corollary 3.2 If graph G contains no H -factors, then there exists two disjoint subsets
f
S,T of V(G) such that
f(S)−|T|+XdG−S(x)−q(S,T) < 0,
x∈T
where q(S,T) denote the number of components C of G − S − T such that |V(C)| +
e (V(C),T) ≡ 1 (mod 2).
G
Proof of Theorem 1.6. Since |V(G)| is even, by Theorem 3.1 (b), G is not H -critical.
f
By Lemma 3.1 (a) and (c), E (B ,C ) = ∅ and every component of G[D ∪B ] is
G H H H H
f f f f
an odd component. We write ω(G[D ∪B ]) = k and G[D ∪B ] = R ∪...∪R .
Hf Hf Hf Hf 1 k
Without loss of generality, suppose that V(R )= V(C )∪···∪V(C )∪B for 1 ≤ i ≤ k,
i i1 iri i
where C is a component of G[D ] for 1 ≤ j ≤ r and B ⊆ B . By Theorem 2.6,
ij H i i H
f f
Theorem 3.1 (c) and Claim 1 of Theorem 3.1,
0 < ∇(Hf)= ω(G[DHf])+|BHf|− X dG−AHf(x)−f(AHf)
x∈BHf
k
= X(|Bi|+ri− X dG−AHf(x))−f(AHf)
i=1 x∈Bi
k
= X(|Bi|+ri−eG(Bi,V(Ri)−Bi))−f(AH )
f
i=1
= k−f(A )
H
f
= o(G[D ∪B ])−f(A )
H H H
f f f
≤ o(G−A )−f(A ),
H H
f f
a contradiction. This completes the proof. 2
Proof of Theorem1.7. ByLemma2.8,GisnotH -critical. ThenwehaveA ∪B 6= ∅.
f Hf Hf
For every component C of G[D ] and every vertex v, we claim E (C ,v) ≤ 1. Otherwise,
i H G i
f
suppose that e (C ,v) ≥ 2. By Lemma 2.7, we have d (v) ≥ 1 for any H -optimal graph
G i F f
F, contradicting to v ∈B .
H
f
10