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Construction of a family of graphs with a small induced proper subgraph with minimum degree 3
Discrete Mathematics 307 (2007) 1999 – 2001 www.elsevier.com/locate/disc
Construction of a family of graphs with a small induced proper subgraph with minimum degree 3 Sul-young Choia , Puhua Guanb,1 a Department of Mathematics, Le Moyne College, Syracuse, NY, USA b Department of Mathematics and Computer Science, University of Puerto Rico, Rio Piedras, Puerto Rico
Received 8 September 2003; received in revised form 25 October 2004; accepted 12 December 2005 Available online 4 December 2006
Abstract We investigate the following question proposed by Erd˝os: Is there a constant c such that, for each n, if G is a graph with n vertices, 2n − 1 edges, and (G) 3, then G contains an induced proper subgraph H with at least cn vertices and (H ) 3? Previously we showed that there exists √ no such constant c by constructing a family of graphs whose induced proper subgraph with minimum degree 3 contains at most n vertices. In this paper we present a construction of a family of graphs whose largest induced proper subgraph with minimum degree 3 is K4 . Also a similar construction of a graph with n vertices and n + edges is given. © 2006 Elsevier B.V. All rights reserved. Keywords: Erdös; Minimum degree; Induced subgraph
1. Introduction There have been many studies on the conditions of a graph which contains or does not contain a subgraph with certain properties, such as cycles or complete graphs. Bollobás presented many such results in his book [1]. Ramsey- and Turántype problems are wellknown among such studies [5,7,4]. Recently, Simonovits and Sós presented an excellent survey on Ramsey- and Turán-type problems with some applications [6]. In this paper we investigate graphs whose induced subgraphs have minimum degree 3. We will assume a graph G is a simple connected graph with a vertex set V (G) and an edge set E(G). The set of graphs with n vertices, e edges, and minimum degree is denoted by G(n, e, ). For the most part, our notation and terminology follow those of Bondy and Murty [2]. Consider a wheel on n ( 5) vertices. Its minimum degree is 3; however, it does not contain an induced proper subgraph with minimum degree 3. When an edge connecting two vertices on its ‘outer circle’ is added, this new graph contains an induced proper subgraph with at least n2 vertices whose minimum degree is 3 (Fig. 1). This observation led Erd˝os to propose the following question: Is there a constant c such that, for each n, if G is a graph with n vertices, 2n − 1 edges, and (G)3, then G contains an induced proper subgraph H with at least cn vertices and (H )3? E-mail addresses: [email protected] (S. Choi), [email protected] (P. Guan). 1 Partially supported by the U.S. Army Research Office Grant DAAD19-00-0152.
0012-365X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2005.12.046
2000
S. Choi, P. Guan / Discrete Mathematics 307 (2007) 1999 – 2001
Fig. 1. n vertices, 2n − 2 edges, n vertices, 2n − 1 edges.
xn-9 y3
y4
xn-10
y8
y7
y5
y6
xn-8
x2 x
y
y
Fig. 2. A graph in G(n, 2n − 1, 3).
Previously Choi and Guan showed that there exists no such constant c for n 64 by √constructing a graph in G(n, 2n− 1, 3) whose induced proper subgraph with minimum degree 3 contains at most n vertices [3]. In this paper we improve this result: for each n8, we construct a graph in G(n, 2n − 1, 3) which contains only one induced proper subgraph with minimum degree 3, K4 . Also, a similar construction for a graph in G(n, n + , 3) is given. 2. Construction Let a fan Fm = F (x1 , x2 , . . . , xm ) be a graph with m ( 2) vertices, {x1 , x2 , . . . , xm }, and 2m − 3 edges, {x1 x2 , x2 x3 , . . . , xm−2 xm−1 , xm x1 , xm x2 , . . . , xm xm−1 }; a line lm = l(x1 , x2 , . . . , xm ) a graph with m vertices, {x1 , x2 , . . . , xm }, and m−1 edges, {x1 x2 , x2 x3 , . . . , xm−1 xm }; and Km =K(x1 , x2 , . . . , xm ) a complete graph on m vertices {x1 , x2 , . . . , xm }. 2.1. Construction of a graph in G(n, 2n − 1, 3) For an integer n (10), define a line A, a complete graph B, and a fan C as follows: A = l4 = l(y1 , y2 , y3 , y4 ), B = K4 = K(y5 , y6 , y7 , y8 ), and C = Fn−8 = F (x1 , x2 , . . . , xn−8 ). Let G be a graph obtained from A, B, and C such that the set of vertices of G is V (A) ∪ V (B) ∪ V (C), and the set of edges of G is E(A) ∪ E(B) ∪ E(C) ∪ {x1 y1 , xn−9 y4 , xn−8 y4 , xn−8 y8 , xn−8 y5 , y1 y5 , y2 y6 , y3 y7 , y4 y8 }. Then G is a graph in G(n, 2n − 1, 3) (Fig. 2). Let us show that B (=K4 ) is the only induced proper subgraph of G with minimum degree 3. Suppose that H is an induced proper subgraph of G with (H )3. If H contains a vertex xi for i = 1, 2, . . . , n − 9 (or a vertex yj for j = 1, 2, 3), then H also contains all three adjacent vertices of xi (or yj ) since (H ) 3. This implies that H must contain all the vertices of G. Similarly, if H contains either y4 or xn−8 , we can show that H must contain all the vertices of G. Therefore neither xi ’s (i = 1, 2, . . . , n − 8) nor yj ’s (j = 1, 2, . . . , 8) are contained in H , and the only induced proper subgraph of G with minimum degree 3 is B. Notice that the same result holds for n = 8 and 9 with a ‘degenerate’ fan—an empty graph or a graph with only one vertex.
S. Choi, P. Guan / Discrete Mathematics 307 (2007) 1999 – 2001
2001
2.2. Construction of a graph in G(n, n + , 3) For an integer n 10, let and be integers satisfying 2, 0 < n, and n2 n + . We will apply a similar construction√to obtain a graph in G(n, n + , 3) whose induced proper subgraph with minimum degree 3 contains at most O( n) vertices. (More precisely, its induced proper subgraph with minimum degree 3 contains at most √ 5+ 8(−2)n+8+9 vertices.) 2 For a given integer n 10, choose the smallest positive integer m satisfying that m 4 and m2 5 − m + (2n + 2)n + . 2 2 √ 5+ 8(−2)n+8+9 . Let A=lm =l(y1 , y2 , . . . , ym ), B =Km =K(ym+1 , ym+2 , . . . , y2m ), and C =Fn−2m = Then m= 2 F (x1 , x2 , . . . , xn−2m ). Since a smallest graph with m vertices and minimum degree 3 contains 3m edges, as many 2 3m m(m−1) − 2 edges can be deleted from B while maintaining the same set of vertices and the minimum degree as 2
of 3. Let us delete m2 − 25 m + (2n + 2) − (n + ) edges from B so that the resulting subgraph has m vertices and minimum degree 3. Call the resulting subgraph D. Then D has 2m − (2n + 2) + (n + ) edges. Construct a graph G from A, C, and D: the set of vertices of G is V (A) ∪ V (C) ∪ V (D), and the set of edges of G is E(A)∪E(C)∪E(D)∪{x1 y1 , xn−2m−1 ym , xn−2m y1 , xn−2m ym , xn−2m ym+1 , xn−2m y2m , y1 ym+1 , y2 ym+2 , y3 ym+3 , . . . , ym y2m }. Then G is a graph with n vertices, n + edges, and minimum degree 3, i.e., G ∈ G(n, n + , 3). As in the previous construction, an induced proper subgraph of G with minimum degree 3 contains neither xi ’s (i = 1, 2, . . . , n − 2m) nor yj ’s (j= 1, 2, . . . , m). Therefore, the largest proper induced subgraph of G with minimum degree 3 is D √ 5+ 8(−2)n+8+9 which contains vertices. 2 2
References [1] B. Bollobás, Extremal Graph Theory, Academic Press, London, 1978. [2] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, The MacMillan Press, London, 1977. [3] S. Choi, P. Guan, On an Erd˝os question concerning the existence of a large proper subgraph with vertices of degree at least 3, Congr. Numer. 151 (2001) 167–171. [4] P. Erdös, Collected Papers of Paul Turán, vols. 1–3, Akadémiai Kiadó, Budapest, 1989 (in Hungarian; see also ce:cross-ref[7] in English). [5] F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. Second Ser. 30 (1930) 264–286. [6] M. Simonovits, V.T. Sós, Ramsey–Turán theory, Discrete Math. 229 (2001) 293–340. [7] P. Turán, On an extremal problem in graph theory, Mat. Lapok 48 (1941) 436–452.
Construction of a family of graphs with a small induced proper subgraph with minimum degree 3 Sul-young Choia , Puhua Guanb,1 a Department of Mathematics, Le Moyne College, Syracuse, NY, USA b Department of Mathematics and Computer Science, University of Puerto Rico, Rio Piedras, Puerto Rico
Received 8 September 2003; received in revised form 25 October 2004; accepted 12 December 2005 Available online 4 December 2006
Abstract We investigate the following question proposed by Erd˝os: Is there a constant c such that, for each n, if G is a graph with n vertices, 2n − 1 edges, and (G) 3, then G contains an induced proper subgraph H with at least cn vertices and (H ) 3? Previously we showed that there exists √ no such constant c by constructing a family of graphs whose induced proper subgraph with minimum degree 3 contains at most n vertices. In this paper we present a construction of a family of graphs whose largest induced proper subgraph with minimum degree 3 is K4 . Also a similar construction of a graph with n vertices and n + edges is given. © 2006 Elsevier B.V. All rights reserved. Keywords: Erdös; Minimum degree; Induced subgraph
1. Introduction There have been many studies on the conditions of a graph which contains or does not contain a subgraph with certain properties, such as cycles or complete graphs. Bollobás presented many such results in his book [1]. Ramsey- and Turántype problems are wellknown among such studies [5,7,4]. Recently, Simonovits and Sós presented an excellent survey on Ramsey- and Turán-type problems with some applications [6]. In this paper we investigate graphs whose induced subgraphs have minimum degree 3. We will assume a graph G is a simple connected graph with a vertex set V (G) and an edge set E(G). The set of graphs with n vertices, e edges, and minimum degree is denoted by G(n, e, ). For the most part, our notation and terminology follow those of Bondy and Murty [2]. Consider a wheel on n ( 5) vertices. Its minimum degree is 3; however, it does not contain an induced proper subgraph with minimum degree 3. When an edge connecting two vertices on its ‘outer circle’ is added, this new graph contains an induced proper subgraph with at least n2 vertices whose minimum degree is 3 (Fig. 1). This observation led Erd˝os to propose the following question: Is there a constant c such that, for each n, if G is a graph with n vertices, 2n − 1 edges, and (G)3, then G contains an induced proper subgraph H with at least cn vertices and (H )3? E-mail addresses: [email protected] (S. Choi), [email protected] (P. Guan). 1 Partially supported by the U.S. Army Research Office Grant DAAD19-00-0152.
0012-365X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2005.12.046
2000
S. Choi, P. Guan / Discrete Mathematics 307 (2007) 1999 – 2001
Fig. 1. n vertices, 2n − 2 edges, n vertices, 2n − 1 edges.
xn-9 y3
y4
xn-10
y8
y7
y5
y6
xn-8
x2 x
y
y
Fig. 2. A graph in G(n, 2n − 1, 3).
Previously Choi and Guan showed that there exists no such constant c for n 64 by √constructing a graph in G(n, 2n− 1, 3) whose induced proper subgraph with minimum degree 3 contains at most n vertices [3]. In this paper we improve this result: for each n8, we construct a graph in G(n, 2n − 1, 3) which contains only one induced proper subgraph with minimum degree 3, K4 . Also, a similar construction for a graph in G(n, n + , 3) is given. 2. Construction Let a fan Fm = F (x1 , x2 , . . . , xm ) be a graph with m ( 2) vertices, {x1 , x2 , . . . , xm }, and 2m − 3 edges, {x1 x2 , x2 x3 , . . . , xm−2 xm−1 , xm x1 , xm x2 , . . . , xm xm−1 }; a line lm = l(x1 , x2 , . . . , xm ) a graph with m vertices, {x1 , x2 , . . . , xm }, and m−1 edges, {x1 x2 , x2 x3 , . . . , xm−1 xm }; and Km =K(x1 , x2 , . . . , xm ) a complete graph on m vertices {x1 , x2 , . . . , xm }. 2.1. Construction of a graph in G(n, 2n − 1, 3) For an integer n (10), define a line A, a complete graph B, and a fan C as follows: A = l4 = l(y1 , y2 , y3 , y4 ), B = K4 = K(y5 , y6 , y7 , y8 ), and C = Fn−8 = F (x1 , x2 , . . . , xn−8 ). Let G be a graph obtained from A, B, and C such that the set of vertices of G is V (A) ∪ V (B) ∪ V (C), and the set of edges of G is E(A) ∪ E(B) ∪ E(C) ∪ {x1 y1 , xn−9 y4 , xn−8 y4 , xn−8 y8 , xn−8 y5 , y1 y5 , y2 y6 , y3 y7 , y4 y8 }. Then G is a graph in G(n, 2n − 1, 3) (Fig. 2). Let us show that B (=K4 ) is the only induced proper subgraph of G with minimum degree 3. Suppose that H is an induced proper subgraph of G with (H )3. If H contains a vertex xi for i = 1, 2, . . . , n − 9 (or a vertex yj for j = 1, 2, 3), then H also contains all three adjacent vertices of xi (or yj ) since (H ) 3. This implies that H must contain all the vertices of G. Similarly, if H contains either y4 or xn−8 , we can show that H must contain all the vertices of G. Therefore neither xi ’s (i = 1, 2, . . . , n − 8) nor yj ’s (j = 1, 2, . . . , 8) are contained in H , and the only induced proper subgraph of G with minimum degree 3 is B. Notice that the same result holds for n = 8 and 9 with a ‘degenerate’ fan—an empty graph or a graph with only one vertex.
S. Choi, P. Guan / Discrete Mathematics 307 (2007) 1999 – 2001
2001
2.2. Construction of a graph in G(n, n + , 3) For an integer n 10, let and be integers satisfying 2, 0 < n, and n2 n + . We will apply a similar construction√to obtain a graph in G(n, n + , 3) whose induced proper subgraph with minimum degree 3 contains at most O( n) vertices. (More precisely, its induced proper subgraph with minimum degree 3 contains at most √ 5+ 8(−2)n+8+9 vertices.) 2 For a given integer n 10, choose the smallest positive integer m satisfying that m 4 and m2 5 − m + (2n + 2)n + . 2 2 √ 5+ 8(−2)n+8+9 . Let A=lm =l(y1 , y2 , . . . , ym ), B =Km =K(ym+1 , ym+2 , . . . , y2m ), and C =Fn−2m = Then m= 2 F (x1 , x2 , . . . , xn−2m ). Since a smallest graph with m vertices and minimum degree 3 contains 3m edges, as many 2 3m m(m−1) − 2 edges can be deleted from B while maintaining the same set of vertices and the minimum degree as 2
of 3. Let us delete m2 − 25 m + (2n + 2) − (n + ) edges from B so that the resulting subgraph has m vertices and minimum degree 3. Call the resulting subgraph D. Then D has 2m − (2n + 2) + (n + ) edges. Construct a graph G from A, C, and D: the set of vertices of G is V (A) ∪ V (C) ∪ V (D), and the set of edges of G is E(A)∪E(C)∪E(D)∪{x1 y1 , xn−2m−1 ym , xn−2m y1 , xn−2m ym , xn−2m ym+1 , xn−2m y2m , y1 ym+1 , y2 ym+2 , y3 ym+3 , . . . , ym y2m }. Then G is a graph with n vertices, n + edges, and minimum degree 3, i.e., G ∈ G(n, n + , 3). As in the previous construction, an induced proper subgraph of G with minimum degree 3 contains neither xi ’s (i = 1, 2, . . . , n − 2m) nor yj ’s (j= 1, 2, . . . , m). Therefore, the largest proper induced subgraph of G with minimum degree 3 is D √ 5+ 8(−2)n+8+9 which contains vertices. 2 2
References [1] B. Bollobás, Extremal Graph Theory, Academic Press, London, 1978. [2] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, The MacMillan Press, London, 1977. [3] S. Choi, P. Guan, On an Erd˝os question concerning the existence of a large proper subgraph with vertices of degree at least 3, Congr. Numer. 151 (2001) 167–171. [4] P. Erdös, Collected Papers of Paul Turán, vols. 1–3, Akadémiai Kiadó, Budapest, 1989 (in Hungarian; see also ce:cross-ref[7] in English). [5] F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. Second Ser. 30 (1930) 264–286. [6] M. Simonovits, V.T. Sós, Ramsey–Turán theory, Discrete Math. 229 (2001) 293–340. [7] P. Turán, On an extremal problem in graph theory, Mat. Lapok 48 (1941) 436–452.