Long induced paths and cycles in Kneser graphs - Semantic Scholar
Graphs and Combinatorics 5, 303-306 (1989)
Graphsand Combinatorics © Springer-Verlag 1989
Long Induced Paths and Cycles in Kneser Graphs Peter Alles 1 and Svatopluk Poljak 2 1 GZS-ZSE, Theodor-Hauss-Allee80, 6000 Frankfurt 90, Federal Republic of Germany 2 Charles University,Dept. of Appl. Math., Malostransk6nfim.25, 11800Praha 1, Czechoslovakia
Abstract. We present some lower and upper bounds on the length of the maximum induced paths
and cycles in Kneser graphs.
The Kneser graph K(n, r) is the graph whose vertex set is the family of all r-element subsets of { 1, 2..... n}, and a pair of vertices forms an edge, ff the corresponding subsets are disjoint. Kneser graphs are often used in order to express some properties of a family of sets of fixed size. Some results on Kneser graphs may be found e.g. in the following papers [1, 2, 3, 4, 5]. Here We propose the following question. What is the maximum possible length of an induced path (resp. cycle) in a Kneser graph? Let us denoted by p(n, r) and c(n, r) the maximum length of an induced path and cycle in K(n, r), respectively. (We define the length as the number of edges.) Let us observe that, f o r t fixed, both p(n, r) and c(n, r) are bounded, since the size of
an induced matching in K(n,r) is at most (2;) by a result ofBollobfis [1]. Hence, we may define functions p(r) and c(r) by
p(r) = max p(n,r) n
and
c(r) = max c(n,r). n
In other words, p(r) is the maximum length of a sequence So ..... Sp~,) of r-element sets such that Si-1 fq Si = ~ and Si fq Si # ~ otherwise. The functions p(r) and c(r) are closely related, since p(r) > c(r)- 2 and c(r) >_ 2p(r - l). To show the latter, we need the following lemma. The (categorical) product G1 × G2 of graphs Gi = (Vi, Ei), i = 1, 2, is the graph with vertex set V1 x V2 and edge set {((xl,x2),(yl,yz))[xiy 1 ~ E 1 and x2y 2 6 E2}. Lemma 1. If Gi is an induced subgraph of K(ni, ri), i = 1, 2, then G 1 × G~ is an induced subgraph of K(n i + n2, r i + r2). [] Observe that C2k can be embedded in Pk × K3 (see Fig. 1), and the inequality c(r) > 2p(r - 1) follows. We present upper and lower bounds on p(r)in the following theorem. The upper bound has also been observed by Z. Tuza.
The length p(r) of the maximum induced path in K(n, r), n sufficiently large, satisfies
Theorem 2.
304
P. Alles, S. Poljak
Fig. 1. Embedding of C,2 in P6 × K3
,i, (ii) p(r) _> 1.2 × 2.5' for r > 3. Proof. (i) We use the following result by Frankl [2]. Let A 1. . . . , Am and B1, ..., B,, be r-sets satisfying A i f-l Bi = ~ for every i = 1. . . . . m, and A i fq Bj # ~ for every
Let S o , . . . , S,, be a path of length m in K(n,r). Set Ai = Si-1 and Bi = Si, i = 1, ..., m, and apply the Frankl's theorem. (ii) We will use Lemma 1 once more. The bound is valid for r = 3, 4, 5 and 6, since P 4 c K ( 5 , 2 ) , P19cK(7,3), P62cK(9,4), and e l s o c K ( l l , 5 ) ( s e e the computational results below). We employ also C6t c K(9, 4). (We use the notation G c H if G is an induced subgraph of H.) For r > 6, we proceed by induction using the following claim. Claim. The product C m x Pat-2, m odd, contains an induced path of length t ( 2 m - 3) + 2. The construction is indicated in Fig. 2. By the claim, C,, x Pk, m odd, contains a path of length at least ½k(2m - 3) + 2. Now, apply the claim to C61 c U(9, 4). Hence p(r) grows with ratio at least x~/(2 x 61 - 3)/3 ~ 2.5. []
/ \ Fig. 2. Embedding of Pk(2rn-3)+2 in Cmx Pak-2 (m = 7, k = 4)
Long Induced Paths and Cyclesin Kneser Graphs
305
Theorem 2 provides an encoding of exponentially large paths in K(n, r) where n/r = 9/4. We show that exponentially large paths and cycles occur already in K(2r + 1,r), though the "product construction" of Lemma 1 cannot be applied here. Theorem 3. We have
c ( 2 r + 1,r)_> 15.2r-2
for r > 4.
Proof. The statement follows from the fact that C6o c K(9, 4), and the following claim. Claim. I f C2k c K(n, r), k even, then C4k c K(n + 2, r + 1).
Proof. Let k = 2t and X 1. . . . . Xk, Y1 .... , Yk be the vertices of C~k with X i N Y1 = iffeither i = j or i = j + 1 or (i, j) = (1, k). Let a, b be elements not in the underlying set of K(n, r), and X~ .... , X'2k, Y~ . . . . . Y~k be the vertices of C4k defined by X~i_3 = X2,_ 1 U {a}, X41_2
=
X2i_ 1 U {b},
X'4i-1 = X2i U {b}, X'4, = X2 , U {a},
Y,~,-3 = Y2i-1 U {f(2i)},
"YJ.,-2 = Yzi-1 U {a}, Y~.i-1 = Y2i 1.3{/(2i + 1)}, Y•, = Y2, U {b},
for i = 1. . . . . t, where f ( j ) ~ Xj\Xj_I. Obviously, X~ N Xj ~ ~ , Y~' ~ Y/ ¢ ~ , X;nY/=~, X ~ + ~ f q Y / = ~ , X ~ f q Y ~ k = Z J and X ~ f q Y j ' ¢ ~ for i > _ j + 3 cyclically. Since X'2i+2 and Y~ are obtained from Xi+~, Y/by adding the same new vertex, and f(i) e X~kX~_~, f(i) ¢ Y~_~, we have X;+ 2 A Y~' ¢ ~ for all i. [] We have some further constructions in addition to those presented in Theorems 2 and 3. A direct construction of a long induced cycle is given by Fig. 3, where Ck~2,,-4)+,~ is embedded into the product of Pak-2 and C,,, rn odd.
Fig. 3. Embedding of Ck(2m_4j+4in C,, x P3~-2(m = 7, k = 4)
306
P. Alles, S. Poljak
Remark. Let us denote by n = n(r) the smallest n for which K(n, r) already contains an induced path of length p(r), i.e. the maximum possible for a given r. We are not able to find any reasonable bound on n(r). It may even be possible that n(r) = 2r + 1. Some computational results. It is easy to see that K(5, 2) (the Petersen graph) contains an induced path of length 4 and cycles of lengths 5 and 6. We have found the following path P19 in K(7, 3) 123
456
127
345
167
234
156
134
256
147
236
145
367
125,
247
135
246
357
146
257
and checked with the help of computer that it is maximum. The lengths of induced cycles in K(7,3) are 6, 7, 8, 9, 10, 12, 13, 16 and 18. Already in K(9, 4) we were not able to complete the search. The longest constructed path was of length 62, and cycles of the following lengths were indicated: 6, 12, 14, 20, 21, 25-61. We have also found an induced path P150 in K(11, 5).
Short induced cycles in K(n, r). In contrary to the maximum cycles, the minimum (induced) cycles in Kneser graphs can be determined. The shortest induced cycles are C6 and Ca in K(2r + 1, r) and K(2r + 2, r), respectively. The length of shortest odd cycle in K(n, r) was determined in [5] as 2[r/(n - 2r)] + 1, where Ix] denotes the smallest integer greater or equal x. Note added in proof. We have been informed that Y. KOHAYAKAWA obtained some similar results as part of his Ph.D. Thesis, see [6].
References 1. Bollob~is,B.: On generalizedgraphs. Acta Math. Acad. Sci. Hung. 16, 447-452 (1965) 2. Frankl, P.: An Extremal Problem for two Families of Sets. Europ. J. Comb. 3, 125-127 (1982) 3. Frankl, P., Furedi, Z.: Extremal problems concerning Kneser graphs. J. Comb. Theory (B) 40, 270-284 (1986) 4. Lov~sz,L.: Kneser's conjecture, chromatic number and homotopy. J. Comb. Theory (A) 25, 319-324 (1978) 5. Poljak, S., Tuza, Z.: Maximum bipartite subgraphs of Kneser graphs. Graphs and Combinatorics 3, 191-199 (1987) 6. Kohayakawa, Y.: A note on Induced Cyclesin Kneser graphs, to appear.
Received: October 1, 1988
Graphsand Combinatorics © Springer-Verlag 1989
Long Induced Paths and Cycles in Kneser Graphs Peter Alles 1 and Svatopluk Poljak 2 1 GZS-ZSE, Theodor-Hauss-Allee80, 6000 Frankfurt 90, Federal Republic of Germany 2 Charles University,Dept. of Appl. Math., Malostransk6nfim.25, 11800Praha 1, Czechoslovakia
Abstract. We present some lower and upper bounds on the length of the maximum induced paths
and cycles in Kneser graphs.
The Kneser graph K(n, r) is the graph whose vertex set is the family of all r-element subsets of { 1, 2..... n}, and a pair of vertices forms an edge, ff the corresponding subsets are disjoint. Kneser graphs are often used in order to express some properties of a family of sets of fixed size. Some results on Kneser graphs may be found e.g. in the following papers [1, 2, 3, 4, 5]. Here We propose the following question. What is the maximum possible length of an induced path (resp. cycle) in a Kneser graph? Let us denoted by p(n, r) and c(n, r) the maximum length of an induced path and cycle in K(n, r), respectively. (We define the length as the number of edges.) Let us observe that, f o r t fixed, both p(n, r) and c(n, r) are bounded, since the size of
an induced matching in K(n,r) is at most (2;) by a result ofBollobfis [1]. Hence, we may define functions p(r) and c(r) by
p(r) = max p(n,r) n
and
c(r) = max c(n,r). n
In other words, p(r) is the maximum length of a sequence So ..... Sp~,) of r-element sets such that Si-1 fq Si = ~ and Si fq Si # ~ otherwise. The functions p(r) and c(r) are closely related, since p(r) > c(r)- 2 and c(r) >_ 2p(r - l). To show the latter, we need the following lemma. The (categorical) product G1 × G2 of graphs Gi = (Vi, Ei), i = 1, 2, is the graph with vertex set V1 x V2 and edge set {((xl,x2),(yl,yz))[xiy 1 ~ E 1 and x2y 2 6 E2}. Lemma 1. If Gi is an induced subgraph of K(ni, ri), i = 1, 2, then G 1 × G~ is an induced subgraph of K(n i + n2, r i + r2). [] Observe that C2k can be embedded in Pk × K3 (see Fig. 1), and the inequality c(r) > 2p(r - 1) follows. We present upper and lower bounds on p(r)in the following theorem. The upper bound has also been observed by Z. Tuza.
The length p(r) of the maximum induced path in K(n, r), n sufficiently large, satisfies
Theorem 2.
304
P. Alles, S. Poljak
Fig. 1. Embedding of C,2 in P6 × K3
,i, (ii) p(r) _> 1.2 × 2.5' for r > 3. Proof. (i) We use the following result by Frankl [2]. Let A 1. . . . , Am and B1, ..., B,, be r-sets satisfying A i f-l Bi = ~ for every i = 1. . . . . m, and A i fq Bj # ~ for every
Let S o , . . . , S,, be a path of length m in K(n,r). Set Ai = Si-1 and Bi = Si, i = 1, ..., m, and apply the Frankl's theorem. (ii) We will use Lemma 1 once more. The bound is valid for r = 3, 4, 5 and 6, since P 4 c K ( 5 , 2 ) , P19cK(7,3), P62cK(9,4), and e l s o c K ( l l , 5 ) ( s e e the computational results below). We employ also C6t c K(9, 4). (We use the notation G c H if G is an induced subgraph of H.) For r > 6, we proceed by induction using the following claim. Claim. The product C m x Pat-2, m odd, contains an induced path of length t ( 2 m - 3) + 2. The construction is indicated in Fig. 2. By the claim, C,, x Pk, m odd, contains a path of length at least ½k(2m - 3) + 2. Now, apply the claim to C61 c U(9, 4). Hence p(r) grows with ratio at least x~/(2 x 61 - 3)/3 ~ 2.5. []
/ \ Fig. 2. Embedding of Pk(2rn-3)+2 in Cmx Pak-2 (m = 7, k = 4)
Long Induced Paths and Cyclesin Kneser Graphs
305
Theorem 2 provides an encoding of exponentially large paths in K(n, r) where n/r = 9/4. We show that exponentially large paths and cycles occur already in K(2r + 1,r), though the "product construction" of Lemma 1 cannot be applied here. Theorem 3. We have
c ( 2 r + 1,r)_> 15.2r-2
for r > 4.
Proof. The statement follows from the fact that C6o c K(9, 4), and the following claim. Claim. I f C2k c K(n, r), k even, then C4k c K(n + 2, r + 1).
Proof. Let k = 2t and X 1. . . . . Xk, Y1 .... , Yk be the vertices of C~k with X i N Y1 = iffeither i = j or i = j + 1 or (i, j) = (1, k). Let a, b be elements not in the underlying set of K(n, r), and X~ .... , X'2k, Y~ . . . . . Y~k be the vertices of C4k defined by X~i_3 = X2,_ 1 U {a}, X41_2
=
X2i_ 1 U {b},
X'4i-1 = X2i U {b}, X'4, = X2 , U {a},
Y,~,-3 = Y2i-1 U {f(2i)},
"YJ.,-2 = Yzi-1 U {a}, Y~.i-1 = Y2i 1.3{/(2i + 1)}, Y•, = Y2, U {b},
for i = 1. . . . . t, where f ( j ) ~ Xj\Xj_I. Obviously, X~ N Xj ~ ~ , Y~' ~ Y/ ¢ ~ , X;nY/=~, X ~ + ~ f q Y / = ~ , X ~ f q Y ~ k = Z J and X ~ f q Y j ' ¢ ~ for i > _ j + 3 cyclically. Since X'2i+2 and Y~ are obtained from Xi+~, Y/by adding the same new vertex, and f(i) e X~kX~_~, f(i) ¢ Y~_~, we have X;+ 2 A Y~' ¢ ~ for all i. [] We have some further constructions in addition to those presented in Theorems 2 and 3. A direct construction of a long induced cycle is given by Fig. 3, where Ck~2,,-4)+,~ is embedded into the product of Pak-2 and C,,, rn odd.
Fig. 3. Embedding of Ck(2m_4j+4in C,, x P3~-2(m = 7, k = 4)
306
P. Alles, S. Poljak
Remark. Let us denote by n = n(r) the smallest n for which K(n, r) already contains an induced path of length p(r), i.e. the maximum possible for a given r. We are not able to find any reasonable bound on n(r). It may even be possible that n(r) = 2r + 1. Some computational results. It is easy to see that K(5, 2) (the Petersen graph) contains an induced path of length 4 and cycles of lengths 5 and 6. We have found the following path P19 in K(7, 3) 123
456
127
345
167
234
156
134
256
147
236
145
367
125,
247
135
246
357
146
257
and checked with the help of computer that it is maximum. The lengths of induced cycles in K(7,3) are 6, 7, 8, 9, 10, 12, 13, 16 and 18. Already in K(9, 4) we were not able to complete the search. The longest constructed path was of length 62, and cycles of the following lengths were indicated: 6, 12, 14, 20, 21, 25-61. We have also found an induced path P150 in K(11, 5).
Short induced cycles in K(n, r). In contrary to the maximum cycles, the minimum (induced) cycles in Kneser graphs can be determined. The shortest induced cycles are C6 and Ca in K(2r + 1, r) and K(2r + 2, r), respectively. The length of shortest odd cycle in K(n, r) was determined in [5] as 2[r/(n - 2r)] + 1, where Ix] denotes the smallest integer greater or equal x. Note added in proof. We have been informed that Y. KOHAYAKAWA obtained some similar results as part of his Ph.D. Thesis, see [6].
References 1. Bollob~is,B.: On generalizedgraphs. Acta Math. Acad. Sci. Hung. 16, 447-452 (1965) 2. Frankl, P.: An Extremal Problem for two Families of Sets. Europ. J. Comb. 3, 125-127 (1982) 3. Frankl, P., Furedi, Z.: Extremal problems concerning Kneser graphs. J. Comb. Theory (B) 40, 270-284 (1986) 4. Lov~sz,L.: Kneser's conjecture, chromatic number and homotopy. J. Comb. Theory (A) 25, 319-324 (1978) 5. Poljak, S., Tuza, Z.: Maximum bipartite subgraphs of Kneser graphs. Graphs and Combinatorics 3, 191-199 (1987) 6. Kohayakawa, Y.: A note on Induced Cyclesin Kneser graphs, to appear.
Received: October 1, 1988
