Odd Sums of Long Cycles in 2-Connected Graphs
by
Cong Teng
A Thesis Submitted to the Faculty of The Charles E. Schmidt College of Science in Partial Fulfillment of the Requirements of the Degree of Master of Science
Florida Atlantic University Boca Raton, Florida August 1999 Odd Sums of Long Cycles in 2-Connected Graphs by Cong Teng
This thesis was prepared under the direction of the candidate's thesis advisor, Dr. Stephen C. Locke, Department of Mathematics, and has been approved by the members of her supervisory committee. It was submitted to the faculty of The Charles E. Schmidt College of Science and was accepted in partial fulfillment of the requirements for the degree of Master of Science.
SUPERVISORY COMMITTEE: ~c~ tL~~
Chairman, Department of Mathematics
Dean of Graduate Studies and Research
11 Acknowledgement
First, I want to give my deepest thanks to my advisor, Dr. Stephen Locke, for his guidance, inspiration and encouragement through my study and mas ter's thesis process. I would also like to give my high appreciation to the committee member, Dr. Fred Richman, for his advice, strict training in Ab stract Algebra, and especially his attitude toward research and life which has some effects on me. I feel deeply thankful to Dr. Heinrich Niederhausen, who is also in my committee, for his support and kind encouragement. Also
Mr. Volker Leek, I deeply appreciate your help in computer skill for ad justing and formatting this thesis. I would like to thank all the faculty in mathematics department who have directly or undirectly supported my life at FAU. I would like to give special thanks to Dr. Yuan Wang for her warm support both as a teacher and a friend.
lll Abstract
Author: Cong Teng
Title: Odd Sums of Long Cycles in 2-connected graphs
Institution: Florida Atlantic University
Thesis Advisor: Dr. Stephen C. Locke
Degree: Master of Science
Year: 1999
Let G be a 2-connected graph with minimum degree d and with at least d + 2 vertices. Suppose that G is not a cycle. Then there is an odd set of cycles, each with length at least d + 1, such that they sum to zero. If G is also non-hamiltonian or bipartite, cycles of length at least 2d can be used.
lV Table of Contents
1. INTRODUCTION 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
2. NOTATION AND TERMINOLOGY ...... 5
3. 2d-KERNELS OF 2-CONNECTED GRAPHS ...... 7
4. (d + 1)-KERNELS OF 2-CONNECTED GRAPHS ...... 29
REFERENCES 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 31
v Section 1. Introduction
The following two theorems appear in Dirac [7], [8].
Theorem 1.1 (Dirac, [7]) Let G be a 2-connected graph with minimum degree d and at least 2d vertices. Then G contains a cycle of length at least
2d.
Theorem 1.2 (Dirac, [8]) Let G beak-connected graph and X be a set of k vertices of G, k 2: 2. Then G contains a cycle C which contains every vertex of X.
The cycle space of a graph G is the vector space spanned by edge sets of cycles of G over G F2 = { 0, 1}. A set of cycles C is a k-kemel if (i) every cycle inC has length at least k, (ii) ICI is odd, and (iii) every edge of G occurs in an even number of cycles of C. In other words, C is an odd set of cycles, each with length at least k , such that the cycles inC sum to zero. Other standard graph-theoretic terminology will be reviewed in Section 2.
Bondy and Lovasz [5] proved that cycles through a specified set of vertices
1 generate the cycle space of a graph.
Theorem 1.3 ([5]) Let S be a set of k- 1 vertices in a k-connected graph
G. Then the cycles through S generate the cycle space of G
Based on Theorem 1.1 and Theorem 1.3, Bondy proposed the following
conjecture.
Conjecture (Bondy, [3] ) Let G be a 3-connected graph with minimum
degree at least d and at least 2d vertices. Then every cycle is the sum of an
odd number of cycles of length at least 2d - 1.
Locke [11], [13] partially proved this conjecture in the following Theorems.
Theorem 1.4 (Locke, [11]) If G is a non-hamiltonian, 3-connected graph with minimum degree d, then every cycle is the sum of cycles of length at
least 2d- 1.
Theorem 1.5 (Locke, [13]) Let G be a 2-connected non-hamiltonian graph
with minimum degree d. Then G has a 2d-kernel.
2 Theorem 1.4 and Theorem 1.5 combine to yield Theorem 1.6.
Theorem 1.6 If G is a non-hamiltonian, 3-connected graph with minimum degree d, then every cycle is the sum of an odd number of cycles of length at least 2d- 1.
Clearly, if G is (2d- 1)-generated and there are an odd number of cycles summing to zero, each with length at least 2d- 1, then every cycle is the sum of an odd number of cycles, each with length at least 2d- 1.
In [9], [10], Hartman gave the following result.
Theorem 1. 7 (Hartman, [9], [10]) Let G be a 2-connected graph with min imum degree d, where G is not Kd+l if d is odd. Then the cycles of length at least d + 1 generate the cycle space of G.
As an extension to Theorem 1.7, we prove the following two theorems.
These results have been submitted to Discrete Mathematics [14].
Theorem 1.8. Let G be a 2-connected graph with minimum degree d, where G is not Kd+l if dis not odd. Then G has a (d + 1)-kernel.
3 Combining Theorem 1.7 and Theorem 1.8, we obtain the following theo rem.
Theorem 1.9 Let G be a 2-connected graph with rninimillil degree d and at least d + 2 vertices. Then every cycle is the sum of an odd nlUilber of cycles of length at least d + 1.
We also give a result which is a little closer to Bondy's Conjecture.
Theorem 1.10 Let G be a 2-connected hamitonian graph with minimum degree d at least 3. Suppose G contains no cycle of length v( G) - 1 or no cycle of length v( G) - 2. Then G has a 2d-kernel.
4 Section 2. Notation and Terminology
All graphs considered in this thesis are simple graphs. Notation and terminology not defined in this thesis can be found in [6]. Let G be a graph.
We denote by v( G) and c-( G) the number of vertices and the number of edges of G , respectively. The degree of vertex v in G is dc(v) or d(v). 8(G) and
.6.(G) are the minimum and maximum degrees of vertices in G. The length of a path or a cycle is the cardinality of its edge set. We use de( u, v) or d(u, v) for the distance from u to v, and P(u, v) the segment of P from u to v. A graph G is connected if every pair of vertices is connected by a path.
G is called k-connected (for k E N) if IGI 2:: k, and G- X is connected for every set X~ V with lXI < k. If IGI > 1 and G- F is connected for every set F ~ E of fewer than l edges, then G is called l-edge-connected. The cycle space of a graph G is the vector space of edge sets of G over GF(2).
A graph is k-generated if its cycle space is generated by cycles of length at least k. A graph is k-path-connected if, for every pair of vertices x and y, there is an (x, y)-path of length at least k. We call a 2-connected graph a k-generator if G is k-generated and (k- I)-path-connected. For a vertex v,
N(v) is the set of neighbors of v in G. For a vertex v and subset X of G,
Nx(v) is the set of neighbors of v in X, that is, Nx(v) = N(v) n V(X). We set nx(v) = INx(v)l. A cut vertex of G is a vertex x such that G-x has more
5 components than G. A connected graph is separable if it has a cutvertex. A maximal connected subgraph without a cutvertex is called a block. A block of G which contains exactly one cut vertex is called an endblock. An internal vertex of an endblock is a vertex of the block different from its cutvertex. vVhen it is necessary, we will assign a direction to a cycle C. In this case, we use C for the opposite direction. Similarly, P denotes the path P travelled in the reverse direction. C[x, y] is the segment of the cycle from x toy, in the direction consistent with the cyclic ordering on C. For each vertex w of C, let w+ denote the vertex immediately following w on C, and w- the vertex immediately preceding w on C. We use w+q for the qth vertex following w, so w+(q+l) = (w+qt. Similarly, w-q is the qth vertex preceding w . For a graph G and a cycle C in G , we call C subharniltonian if v (C) < v (G).
6 Section 3. 2d-Kernels of 2-Connected Graphs
We begin by displaying two lemmas.
Lemma 3.1 ([16] Exercise 10.19) Let G be a nonseparable graph, let u, v be vertices of G (with u =/: v unless G ~ Kt). Suppose that 6. ~ d and that each vertex of G other than u and v has degree at least d. Then there is a
( u , v )-path in G of length at least d.
For completeness of this thesis and for the convenience of the reader, we include the proof of the following lemma.
Lemma 3.2 (Locke, [13]) If G is a 2-connected non-hamiltonian graph with minimum degree d, then G contains a 2d-kernel.
Proof: Suppose Cis a longest cycle of G. Then C is not a Hamilton cycle.
Let H be a component of G- C.
In [4], it wa.c; proven that there is a (u, v)-path Pin H with the following properties:
(i) There are distinct vertices u' and v' on C, with uu', vv' E E (G); and
(i·i) E (P) ~ d- s, where s = nc (v).
7 For convenience, we give a short recapitulation of that proof. If v (H) =
1, then u = v E V (H), and we choose any two distinct neighbors of v for u' and v'. See Figure 3 .1.
v' u'
v' u'
v u v
Figure 3.1 Figure 3.2
If H is non-separable and v (H) 2: 2, we choose distinct vertices u and v in H so that they each have at least one neighbor on C, they collectively have at least two neighbors on C, and subject to these constraints, we choose v so that nc ( v) = INc (v) I is maximum among vertices of H . Any vertex
8 vo in the subgraph H has degree dH (vo) 2: d- s. By Lemma 3.1, there is a
path P[u, v] in H with length at least d- s. See Figure 3.2.
v' u'
v b u
c
G'
Figure 3.3 Figure 3.4
If H is separable, we choose distinct vertices u and v in H so that they each have at least one neighbor on C, they collectively have at least two neighbors on C, and subject to these constraints, we may choose v to be internal to an endblock B of H, and u not internal to B. Let b be the cutvertex of H which is in B. We may also assume that v has been selected
9 so that nc (v) is maximum among internal vertices of B. Any internal vertex v0 in the block B has degree d8 (v0 ) 2: d- s. By Lemma 3.1, there is a path P'[b, v] in B with length at least d- s, and a path P[u, v] in H with length at least d- s. See Figure 3.3.
Let { u'} U Nc (v) ={xi, x2, . .. , xe}, with u' =XI and with xi, x2, .. . , Xt occnrring in this order cyclically on C. (In all proofs, subscripts are modulo m, for some m; that is Xt+l = x1 here.) Note that s :::; t :::; s + 1. See Figure
3.4.
v u
c~ c,
Figure 3.5 Figure 3.6
10 Let G" = C U {u, v} U P[u, v] U {uxr} U {vxi: i = 2, 3, . .. , t}. Also see
Figure 3.4. We shall show that there is a 2d-kernel in G". For 2 ::; i ::; t- 1, let ci = C[xi+l,xi]XiVXi+l, and let cl = C[x2,xdxluPvx2, and
Ct = C [x 1 , xt] XtvPux 1 . See Figure 3.5, Figure 3.6 and Figure 3.7.
Since Cis a longest cycle of G, E (C [xi, xi+1]) 2: 2, fori= 2, 3, ... , t- 1, and E (C [xi, xi+l]) 2: d-s+2, fori= 1 and i = t. Now for 1 ::; i::; t, E (Ci) 2:
2(d- s + 2) + 2(t- 1) = 2d + 2 (t- s + 1) 2: 2d. Also, E (C) 2: 2d. When t is even, C1 + C2 + · · · + Ct + C = 0. When t is odd, C1 + C2 + · · · + Ct = 0. In both cases, G" has a 2d-kernel.
v u
Ce-
Figure 3.7
11 In the following two theorems, we slighly expand the class of the graphs considered in Lemma 3.2.
Theorem 3.3 Let G be a 2-connected hamiltonian graph with 6 2: d 2: 3.
Suppose that G contains no cycle of length exactly v( G) - 1. Then G has a
2d-kernel.
Proof: We begin with one special case. Suppose that G "' Kd,d and let
C = x 1x 2 ... x 2dx1 be a Hamilton cycle of G. Then, x 1x 4x 5x 2x 1 = C + x 1x 4C [x4, x2] x2xsC[xs,xJ See Figure 3.8. By symmetry, every 4-cycle of G is the sum of two Hamilton cycles. Let Fi = xixi+lx2d-iX2d-i+lxi,
1 s; i s; d- 1. But now, C + F1 + F2 + · · · + Fd-l = 0. See figure 3.9. There fore Kd,d has a 2d-kernel. For the remainder of the proof, we may assume that G ~ Kd,d·
Figure 3.8 Figure 3.9 12 Suppose Cis not a Hamilton cycle of G, but, subject to this, assume that v (C) is maximum. Let H be a component of G - C.
As in the proof of Lemma 3.2, there is a (u , v )-path P in H with the following properties:
(i) There are distinct vertices u' and v' on C , with uu', vv' E E (G); and
(ii) E (P) 2:: d- s, where s = nc (v).
We may also assume that among all candidates for { u, v} satisfying (i) and (ii) , we have
chosen { u, v} such that dH (u, v) is minimum.
Figure 3.10 Figure 3.11
13 Let {u'} U Nc (v) = {x1,x2, ... , xt}, with u' = x1 and with x1, x2,
. . . , Xt occurring in this order cyclically on C, with s ~ t ~ s + 1. See
Figure 3.10. For 2 ~ i ~ t - 1, let ci = c [xi+l ' xi] XiVXi+l ' and let cl = c [x2, xl] xluPvx2, and Ct = c [xl, Xt] XtVPuxl·
We need only show that each Ci has length at least 2d, and that, when t is even, C has length at least 2d. Then, when tis even, C1 +C2 +· · ·+Ct+C = 0, and when tis odd, C1 + C2 + · · · + Ct = 0. In both cases, G has a 2d-kernel.
Figure 3.13 Figure 3.12
14 Suppose that xixi+1 E E (C) for some i, 2 ::; i ::; t - 1. Then, c (Ci) = c (C) + 1 ::; v (G) - 1, contradicting the choice of C . Therefore, we may
assume that c (C [xi, xi+1]) 2: 2, for 2 ::; i ::; t- 1. Note also, if x1 E N (v), then c (C [xi, xi+ 1]) 2: 2, for 1 ::; i ::; t. See Figure 3.11. Without loss of generality, c (C [x1, x2]) ::; c (C [xt, xi]). If x1x2 tt E (C), then c (C) 2: c (Ci)
and, hence, c (C [x 1 , x2]) 2: d-s+2. Therefore, c (Ci) 2: 2d for alli, 1 ::; i ::; t
and c(C) 2: c(C1) 2: 2d.
Figure 3.14 Figure 3.15
15 We must now consider the possibility that xix2 E E (C), xi tf. N (v), and c (P) ~ 1. See Figure 3.12. Then, c (CI) = c (C)+ 1 +c (P) ~ c (C) +2. By the choice of C, v (CI) = v (G). Therefore, V (P) = V (H)= V (G)- V (C).
Let P = YIY2 . . . Ym with YI = u and Ym = v. Suppose that there is an edge y1yk, k ~ j + 2. See Figure 3.13. Let Q = YIY2 . .. Y1Yk . .. Ym and
C~ = C [x2, xi] xiuQvx2. Then, v (G)> c (Ci) = c (C)+1+c (Q) ~ c (C)+2, contradicting the choice of C. Therefore, E (P) = E (H). But now, nc (yi) = nc (u) ~ 2 and nc (y2) ~ d-dH (y2) ~ 1. Therefore, {u, Y2} was an allowable choice when we picked {u, v }. Hence, dH (u, v) = 2, and H ~ K 2. See Figure 3.14.
But now t ~ d. Suppose that c (C [xt, xt]) ~ 2. Then c (C) ;::: c (Ct) and, hence, c (C [xt,xd) ~ 3. But then c (Ci);::: 2 (t- 2) + 3;::: 2d for all i ,
1 ~ i ~ t , and c ( Ci) ;::: 2d. Hence, we may assume that c (C [xt, x1]) = 1.
Also, if t > d, then c(Ci) ;::: 2 (t- 2) + 2 2: 2d for all i, 1 ~ i ~ t, and c: (C) ;::: 2d. Thus, we may assume that .t = d = d (u) = d (v).
Suppose that ux2 E E(G). See Figure 3.15. Then x1ux2C[x2,xr] is a (v(G) -I)-cycle. Therefore, ux2,uxt tf. E(G). Suppose that uxq E E(G), for some q, 2 ~ q ~ t -1. See Figure 3.16. Then c~ = XqUVXq+1C[xq+I,xq] is subhamiltonian and, hence, c (C [xq,Xq+1]) ;::: 3 and, by considering C~-I = xq_ 1vuxqC [xq, Xq_ 1], c (C [xq- 1,xq]) ~ 3. Note that C~+C~_ 1 = C [xq-I , Xq+d Xq+Ivxq,c(Ci) ~ 2d,fori E {1,2, ... ,t}\{q-1,q},c(C~- 1 ) ~ 2d,c(C~) ~ 2d, and c: (C) ;::: 2d. When t is even, C1 + C2 + · · · + Cq-2 + C~_ 1 + C~ +
16 Cq+l + · · · + Ct + C = 0, and when t is odd, C1 + C2 + · · · + Cq_ 2 + C~_ 1 + C~ + Cq+l + · · · + Ct = 0. In both cases, G has a 2d-kernel.
Figure 3.16 Figure 3.17
17 Therefore, we may assume that no neighbor of v is also a neighbor of u, that every neighbor of u is adjacent on C to two neighbors of v, and every neighbor of v is adjacent on C to two neighbors of u. The neighbors of v alternate with the neighbors of u around C. See Figure 3.17. Now suppose that xixj E E(G). Then C [xi ,xj] xjuxjC [xj,xi-l] xi_ 1vx1xi is a ( v (G) - 1 )-cycle. See Figure 3.18. Therefore, no two neighbors of v are adjacent, and similarly, no two neighbors of u are adjacent. Therefore, G is bipartite and G ~ Kd,d·
Figure 3.18 Figure 3.19
18 Corollary Let G be a bipartite 2-connected graph with 8 2: d. Then G has a 2d-kernel.
Proof: If G is non-hamiltonian, use Lemma 3.2. If G is hamiltonian, then
G can have no cycle of length v (G)- 1 . Use Theorem 3.3.
Lemma 3.4 Let G be a 2-connected graph with minimum degree d, and with no (v(G) - m)-cycle, for some m, 2 ~ m ~ d- 1. Then G has a
(2d- m)-kernel.
Proof: By Lemma 3.2, G must have a Hamilton cycle and, by Theorem
3.3, G must have a cycle C of length v (G) - 1. Let N (G) \N (C) = { v}, and let x 1 , x 2 , ... , Xt be the neighbors of v on C , ordered cyclically around
C. See Figure 3.19. Let Ei = E (C [xi, xi+lD· If E (C [xi, xj]) = m + 1, then xivxjC [xj, xi] is a (v (G)- m)-cycle. Therefore, Ei,i = Ei # m + 1,
Ei,i+l = Ei + Ei+l # m + 1, · · · , and Ei,i+m = Ei + Ei+l + · · · + Ei+m # m + 1, for each i, 1 ~ i ~ t. But now, (ci,i, Ei,i+l, ... , Ei,i+m) is a sequence of m + 1 integers, some two of which, say Ei,r and Ei,s must have the same residue modulo m. But then, Er+l,s = Er+l + Er+2 + · · · + Es = O(mod m). If
Er+l,s = m , then G has a (v (G)- m)-cycle. Therefore, Er+l,s 2: 2m. Hence,
Ei,i+m 2: 2 (m + 1) , for all i. Therefore, E (C) 2: 2t 2: 2d, and v (G) 2: 2d + 1.
19 Let ci = XiVXi+lc [xi+l, x;]. Lett= (m + 1) n + j, where 1 ~ j ~ m + 1.
Then, c ( Ci) ~ 1 + j + 2 (m + 1) n = 2t + 1-j ~ 2t- m ~ 2d- m. Therefore, G has a (2d- m)-kernel.
Theorem 3.5 Let G be a 2-connected graph with minimum degree at least d, and with no (v(G)- 2)-cyde. Then G has a 2d-kernel.
Proof: We now consider the case m = 2 in Lemma 3.4. If Ei ~ 2 for all i, then G has a 2d-kernel. Therefore, we may assume that Ej = 1 for some j.
Suppose that Ei = 4, for some i. If xixi+1 E E (G), then xtx;+1C [xi+ 1 , xt] is a (v(G)- 2)-cycle. Therefore, xix;+1 tf. E(G). See Figure 3.20.
D
Figure 3.20 Figure 3.21
20 Let D = xivxi+l C [xi+ 1 , xi]. See figure 3.21. Thus, D is a ( v (G) - 3) cycle with G-V (D)= C [xi , xH- 1] a path oflength two. Let w 1 , w 2 , .. . , Wr be the neighbors of xi on D, ordered cyclically around D. See Figure 3.22.
No two of these vertices may be consecutive on D, since G has no ( v (G) - 2) cycle. Hence, if nD (xt) = r 2:: d, then G has a 2d-kernel. Therefore, we may
2 assume that nD (xt) = d-1 = nD (xH- 1). Now suppose that waxi E E (G)
2 for some o: . If E (D [wa-1, wa]) = 2, then D [wa , Wa-d Wa-1xtxi wa is a
(v (G)- 2)-cycle. See Figure 3.23. Thus, E (D [wa-1, wa]) 2:: 3 and, similarly,
E (D [wa, Wa+1]) 2:: 3. The set of cycles
{ D [wq+1, wq] Wqxtwq+1} qofa ,a-1 U {D , D [wa+1 >wa] Waxi2xiwa+1, D [wa , Wa-1] Wa-lxtxi2wa} contains a 2d-kernel. Therefore, we may assume that waxi2 tf. E (G) for all
0:.
2 Let y E N D ( xi ). Without loss of generality, y E D (Wr , w1 ) . See Figure 2 3.24. Now, if E (D [y , w1]) = 2, then D [w1, y] yxi xiw1 is a (v (G)- 2) - cycle Therefore, E (D [y, w1]) I=- 2 and similarly E (D [wr , y]) I=- 2 . In fact, no element of N D (xi ) can be at distance two on D from a member of N D ( xi2).
Suppose that E (D [y , w1]) 2:: 3 . Then, the set of cycles
{ D [wq+1, wq] Wqxi wq+l} qofr
U {D , D [w1, y] yxi 2xiw1 , D [y , Wr] Wr xixi 2y}
21 contains a 2d-kernel. Therefore, we may assume that E (D [y , w 1]) = 1 and,
2 similarly, E (D [wr, y]) = 1. Every element of Nn (xi ) must be at distance
one on D from two elements of N D (xi) 0 By a similar argument, every
2 element of N D ( xi ) must be at distance one on D from two elements of
2 ND (xi+ 1) and, since nD (xi ) ~ d- 2 = nD (xt) - 1 = nD (xi+I) -- 1,
N D ( xi+1) = N D ( xt) 0 Also, without loss of generality, E (D [wr , w 1]) > 2 and E (D [wq, Wq+l]) = 2, for q =/= 1.
D
Figure 3022 Figure 3023
22 2 2 Let F = D [w1, y] yxT xTw1. Then, Np (xi+1) = {wq}~=l U { xt }. Also, 2 2 E (F [wq ,Wq+l]) 2: 2, for q =/= r , and E (F [wr,xT ]) = 2 = E (F [xt ,wl]). See Figure 3.25. Thus, F is a (v (G)- 1)-cycle and x;+l tl. V (F) and every pair of consecutive neighbors of xi+ 1 on F are at distance at least 2. There fore, G contains a 2d-kernel.
Figure 3.24 Figure 3.25
23 The above argument demonstrates that G cannot have a (v (G)- 3)-cycle C, such that G- v (c) is a path.
We may now assume that £i =J. 4 and £i =J. 3, for all i, and Ej = 1, for some
J. Suppose that £i = 2, for some i. 'vVe may choose i so that £i+l =J. 2, and then £i+ 1 2: 5. Choose (3 such that Ef3-l =J. 2 and Ef3 = Ef3+l = · · · = Ei (we allow (3 = i.). Since £{3_ 1 2: 5, (3- 1 =J. j, i. Hence, (3 tt {i + 1, i + 2, i + 3}. If (3 = i + 4, then£ (Cq) 2: (d- 3) 2 + 2 + 1 + 5 = 2d + 2 and£ (C) 2: 2d + 5. In this case, { C} U { Cq} ~= 1 contains a 2d-kernel. See Figure 3.26. We may therefore assume that (3 =/:. i + 4, and thus d 2: 5.
24 We note that
E[J -3, {3 -1 > 1 + 1 + 5 = 7,
E[J - 2,{3 > 1 + 5 + 2 = 8,
Ei-l,i+l > 2 + 2 + 5 = 9,
Ei,i+2 > 2 + 5 + 1 = 8,
ci+l,i+3 > 5 + 1 + 1 = 7, and
Eq ,q+2 2: 6 for all q. Therefore, 3c (C) 2: (7 + 8 + 9 + 8 + 7) + (d - 5) 6 = 6d + 9 and c (C) ~ 2d + 3. If Eq :::; 2, then c( Cq) = c( C) - Eq + 2 ~ 2d + 3. If Eq ~ 5, we need to be a little more careful. However, in determining the bound on c (C), we never assumed that Eq > 5. That is, one could replace C [xq, Xq+l] by a path P [xq, Xq+d of length five. Then, c ( C [xq+l , xq] P [xq, Xq+l]) 2: 2d + 3 and c (Cq) 2: 2d. We may therefore assume that Ei =/= 2, for all i.
Since Eq,q+2 2: 7, for every q, c (C) 2: ~d . Lett= 3n+s, where 1:::; s:::; 3. Then c(Cq) 2: 7n + 1 + s = (6n + 2s) + (n + 1- s) = 2t + (n + 1- s). If n ;:::: 2 or if s = 1 or if t > d , then c (Cq) 2: 2d. We may therefore assume that t =dE {3, 5, 6}. If d = 5. c (Cq) ~ 5 + 2 + 3 · 1 = 10. Thus, dE {3, 6} .
Suppose d = 3. Relabel C so that C = z1z2 . .. z-yz1, with N (v) =
{z1 , z2, z3} and 1 = v (G) - 1 ~ 7. See Figure 3.27. Since G has no
(v (G) - 2)-cycle, v, z2 , z5, Z7 tf. N (z4 ) and z5z1 tf. E (G). Therefore, z4 has
25 a neighbor on C [z8, z1]. (If 1 = 7, then zs = zi). If z4z8 E E (G), then
D = C [z8, z4] z4z8 is a (v (G)- 3)-cycle with G- V (D) a path of length two.
Thus, G has a (2d)-kernel. Therefore, z4zs t/. E (G) and z4 has a neighbor on
C [z9, zi]. If Zg =/= z1, then {C [z2 , z1] Z1VZ2 , C [z4, Zg] ZgZ4, C [zg, z1] Z1VZ2Z3Z4Z9}
is a 2d-kernel. Thus, we may assume that z9 = z1, and similarly, z8z3 E
E (G) . See Figure 3.28. But, since G has no ( v (G) - 2)-cycle, z1 , z2 , z3 , z7 , z8 , 7! t/.
N (z5 ), leaving d (z5 ) = 2 < d. Therefore, we may now assume that d = 6.
v
Figure 3.28 Figure 3.29
26 If there are more than two choices of q such that Eq 2: 5 , then G has a 2d-kernel. If there are two choices of q such that Eq 2: 6, then G has a
2d-kernel. Thus, we may relabel C so that C = z1z2 .. . z1 z1, with N ( v) =
{z1, z2, z3, z8, z9 , z10 }, and 1 2: 14. See Figure 3.29. Let D = C [z8, z3] z3vz8.
Note that G- V (D) = C [z4 , z7] is a path of length three. Let s 1 , s2 , .. . ,s11 be the neighbors of z4 on D , in this order cyclically on D, with s 11 = z3 , and with the orientation along D agreeing with that along C. See Figure 3.30.
If sisi+l E E (D), for some i, then i5 = D [si+l, si] siz4si+1 is a (v (G)- 3) cycle, with G - V ( i5) = C [z5 , z7] and, therefore, G has a (2d)-kernel.
Hence, no two neighbors of z4 are adjacent on D. If z4z8 E E (G), then c = c [zs, z4] Z4Z8 is a (v (G)- 3) -cycle, with G- v (c) = c [zs, z7] and, therefore, G has a 2d-kernel. Thus, z4z8 ~ E (G). If z4 z9 E E (G) , then
C[z9,z1]z1vzsz7z5zsz4z9 is a (v(G)-2)-cycle. Thus, z4 zg ~ E(G). But now,
contains a 2d-kernel.
27 Figure 3.30 For the class of graphs considered in Theorem 3.3 and Theorem 3.5, if we can prove the cycle space is (2d - 1 )-generated, then by the above two theorems, every cycle is the sum of an odd number of cycles, each of length at least 2d- 1.
28 Section 4. (d + 1)-Kernels of 2-Connected Graphs
Hartman gave the following result [9], [10].
Theorem 4.1 (Hartman [9], [10]) Let G be a 2-connected graph with mini mum degree d, where G is not Kd+I if dis odd. Then the cycles of length at least d + 1 generate the cycle space of G.
We show that if G is a 2-connected graph with minimum degree d and
G ~ Kd+l if dis even, then every cycle of G is the sum of an odd number of cycles, each with length at least d + 1.
Theorem 4.2 Let G be a 2-connected graph with minimum degree d.
Suppose that G is not a cycle and, if dis not even, G is not Kd+l· Then G has a ( d + 1 )-kernel.
Proof. If d = 2, then some two vertices of G are connected by three internally-disjoint paths, P 1, P2 , P3 . But the g U P 2 U P3 contains a 3- kernel. Thus, we may assume that d 2: 3. We may also assume that G has no 2d-kernel. Then, by Theorem 3.3, G has a cycle C of length v (G) - 1.
Let V (G) \V (C) = {v}, and let x1, x2, ... , xk be the neighbors of von
29 C, k ~ d. Let ci = c [xi+l, xi] XiVXi+l· Then E: (Ci) ~ d + 1. If k is even and v (C) ~ d + 1, then C1 + C2 + · · · + Ck + C = 0 . If k is odd, then
C1 + C2 + · · · + Ck = 0. In either case, G has a (d + 1)-kernel.
Theorem 4.1 and Theorem 4.2 together yield the following result. The hypothesis that v ~ d + 2 is necessary. When v = d + 1, G ~ Kd+l · If dis even, there can be no ( d + 1 )-kernel. If d is odd, no odd cycle is the sum of
Hamilton cycles.
Theorem 4.3. Let G be a 2-connected graph with b ~ d, v ~ d + 2. Then every cycle of G is the sum of an odd number of cycles, each of length at least d + 1.
Proof. If G is a cycle, there is nothing to prove. If G is not a cycle, G has a (d + 1)-kernel. Each cycle can represented by a sum of cycles with lengths at least d + 1. If the number of cycles in the sum is even, simply augment this sum by the cycles in a (d +!)-kernel.
The remaining problem is: if G is a 2-connected Hamiltonian graph and has cycles of length exactly v( G) - 1 and v( G) - 2, does G have a (2d- !) kernel? There are some results in [2] about (2d- !)-generated cycle spaces.
30 References
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32