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A remark concerning graphical sequences
Discrete Mathematics 304 (2005) 62 – 64 www.elsevier.com/locate/disc
Note
A remark concerning graphical sequences Geir Dahla , Truls Flatberga, b,∗ a Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway b Department of Informatics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway
Received 14 October 2003; received in revised form 28 September 2005; accepted 29 September 2005 Available online 28 October 2005
Abstract In [A. Tripathi, S. Vijay, A note on a theorem of Erdös & Gallai, Discrete Math. 265 (2003) 417–420] one identifies the nonredundant inequalities in a characterization of graphical sequences. We explain how this result may be obtained directly from a simple geometrical observation involving weak majorization. © 2005 Elsevier B.V. All rights reserved. Keywords: Graphical sequence; Majorization
A sequence of positive integers d1 , d2 , . . . , dp is called graphical if it is the degree sequence of a graph, i.e., there is a graph whose vertices have degrees d1 , d2 , . . . , dp . Sierksma and Hoogeveen [5] presents several (actually seven) equivalent conditions for an integer sequence to be graphical. For a further discussion of this topic, see [3, Chapter 7] or [4, Chapter 3]. One well-known characterization of graphical sequences is the following theorem of Erdös and Gallai [1]. Theorem 1 (Erdös and Gallai [1]). Anonincreasing sequence of positive integers p d1 , d2 , . . . , dp is graphical if and only if i=1 di is even and k i=1
di k(k − 1) +
p
min(k, di )
(k p).
(1)
i=k+1
∗ Corresponding author. Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern,
0316 Oslo, Norway. E-mail address: trulsf@ifi.uio.no (T. Flatberg). 0012-365X/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2005.09.005
G. Dahl, T. Flatberg / Discrete Mathematics 304 (2005) 62 – 64
63
U V
0
1
2
n
Fig. 1. Weak majorization geometrically.
A strengthening of this result was given in [6]. The indices 1 i p − 1 with di > di+1 are denoted by 1 , 2 , . . . , l−1 , and define l = p. Theorem 2 (Tripathi and Vijay [6]). In Theorem 1 it suffices to check the inequalities (1) for k = 1 , 2 , . . . , l . Define K = max{i : di i}. Similar to [6] we observe that it suffices to check the inequalities (1) for k K. (The argument here is similar to the one in [6] except that their maximum k might be one larger than K). Let d ∗ be the conjugate sequence of d with elements given by dk∗ = |{i : di k}| (and with trailing zeros). This is a nonincreasing sequence. Let k K. Then the kth inequality in (1) becomes k i=1
(di + 1)
k
di∗
(k K).
(2)
i=1
These inequalities are referred to as the Hässelbarth criterion in [5] while [3] attributes this result to Ruch and Gutman; see these papers for the appropriate references. Define sequences d and d ∗ with elements dk + 1 and dk∗ for k = 1, . . . , K, respectively. Then condition (2) says that d is weakly majorized by d ∗ , denoted by d ≺w d ∗ . We refer to [2] for a comprehensive treatment of the notion of majorization. Majorization may be interpreted geometrically as follows. Let u be a sequence with nonincreasing elements u1 u2 · · · un . Consider the associated points (k, Uk ) (k = 0, . . . , n) in the plane, where Uk = ki=1 uk for k 1 and U0 = 0. The linear interpolant of these points is called the Lorenz curve associated with u. Since the elements of u are nonincreasing this curve will be concave. If u and v are two nonincreasing sequences, then v is weakly majorized by u if and only if the Lorenz curve of u lies above that of v, see Fig. 1. Due to concavity it suffices to check the majorization condition (Vk Uk ) at the endpoints of the linear segments (i.e., the breakpoints) of the Lorenz curve associated with v. This is our key observation.
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G. Dahl, T. Flatberg / Discrete Mathematics 304 (2005) 62 – 64
We now apply this observation to our original problem and consider the weak majorization k di ) for d ≺w d ∗ . The breakpoints of the Lorenz curve associated with d are (k , i=1 k = 1, 2, . . . , l. Therefore Theorem 2 is an immediate consequence of our key observation. Acknowledgements The authors thank the referees for some useful comments. References [1] P. Erdös, T. Gallai, Graphs with prescribed degree of vertices, Mat. Lapok 11 (1960) 264–274 (in Hungarian). [2] A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, NewYork, 1979. [3] R. Merris, Graph Theory, Wiley, New York, 2001. [4] U. Peled, N. Mahadev, Threshold graphs and related topics, Annals of Discrete Mathematics, vol. 56, NorthHolland, Amsterdam, 1995. [5] G. Sierksma, H. Hoogeveen, Seven criteria for integer sequences being graphic, J. Graph Theory 15 (1991) 223–231. [6] A. Tripathi, S. Vijay, A note on a theorem of Erdös & Gallai, Discrete Math. 265 (2003) 417–420.
Note
A remark concerning graphical sequences Geir Dahla , Truls Flatberga, b,∗ a Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway b Department of Informatics, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway
Received 14 October 2003; received in revised form 28 September 2005; accepted 29 September 2005 Available online 28 October 2005
Abstract In [A. Tripathi, S. Vijay, A note on a theorem of Erdös & Gallai, Discrete Math. 265 (2003) 417–420] one identifies the nonredundant inequalities in a characterization of graphical sequences. We explain how this result may be obtained directly from a simple geometrical observation involving weak majorization. © 2005 Elsevier B.V. All rights reserved. Keywords: Graphical sequence; Majorization
A sequence of positive integers d1 , d2 , . . . , dp is called graphical if it is the degree sequence of a graph, i.e., there is a graph whose vertices have degrees d1 , d2 , . . . , dp . Sierksma and Hoogeveen [5] presents several (actually seven) equivalent conditions for an integer sequence to be graphical. For a further discussion of this topic, see [3, Chapter 7] or [4, Chapter 3]. One well-known characterization of graphical sequences is the following theorem of Erdös and Gallai [1]. Theorem 1 (Erdös and Gallai [1]). Anonincreasing sequence of positive integers p d1 , d2 , . . . , dp is graphical if and only if i=1 di is even and k i=1
di k(k − 1) +
p
min(k, di )
(k p).
(1)
i=k+1
∗ Corresponding author. Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern,
0316 Oslo, Norway. E-mail address: trulsf@ifi.uio.no (T. Flatberg). 0012-365X/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2005.09.005
G. Dahl, T. Flatberg / Discrete Mathematics 304 (2005) 62 – 64
63
U V
0
1
2
n
Fig. 1. Weak majorization geometrically.
A strengthening of this result was given in [6]. The indices 1 i p − 1 with di > di+1 are denoted by 1 , 2 , . . . , l−1 , and define l = p. Theorem 2 (Tripathi and Vijay [6]). In Theorem 1 it suffices to check the inequalities (1) for k = 1 , 2 , . . . , l . Define K = max{i : di i}. Similar to [6] we observe that it suffices to check the inequalities (1) for k K. (The argument here is similar to the one in [6] except that their maximum k might be one larger than K). Let d ∗ be the conjugate sequence of d with elements given by dk∗ = |{i : di k}| (and with trailing zeros). This is a nonincreasing sequence. Let k K. Then the kth inequality in (1) becomes k i=1
(di + 1)
k
di∗
(k K).
(2)
i=1
These inequalities are referred to as the Hässelbarth criterion in [5] while [3] attributes this result to Ruch and Gutman; see these papers for the appropriate references. Define sequences d and d ∗ with elements dk + 1 and dk∗ for k = 1, . . . , K, respectively. Then condition (2) says that d is weakly majorized by d ∗ , denoted by d ≺w d ∗ . We refer to [2] for a comprehensive treatment of the notion of majorization. Majorization may be interpreted geometrically as follows. Let u be a sequence with nonincreasing elements u1 u2 · · · un . Consider the associated points (k, Uk ) (k = 0, . . . , n) in the plane, where Uk = ki=1 uk for k 1 and U0 = 0. The linear interpolant of these points is called the Lorenz curve associated with u. Since the elements of u are nonincreasing this curve will be concave. If u and v are two nonincreasing sequences, then v is weakly majorized by u if and only if the Lorenz curve of u lies above that of v, see Fig. 1. Due to concavity it suffices to check the majorization condition (Vk Uk ) at the endpoints of the linear segments (i.e., the breakpoints) of the Lorenz curve associated with v. This is our key observation.
64
G. Dahl, T. Flatberg / Discrete Mathematics 304 (2005) 62 – 64
We now apply this observation to our original problem and consider the weak majorization k di ) for d ≺w d ∗ . The breakpoints of the Lorenz curve associated with d are (k , i=1 k = 1, 2, . . . , l. Therefore Theorem 2 is an immediate consequence of our key observation. Acknowledgements The authors thank the referees for some useful comments. References [1] P. Erdös, T. Gallai, Graphs with prescribed degree of vertices, Mat. Lapok 11 (1960) 264–274 (in Hungarian). [2] A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, NewYork, 1979. [3] R. Merris, Graph Theory, Wiley, New York, 2001. [4] U. Peled, N. Mahadev, Threshold graphs and related topics, Annals of Discrete Mathematics, vol. 56, NorthHolland, Amsterdam, 1995. [5] G. Sierksma, H. Hoogeveen, Seven criteria for integer sequences being graphic, J. Graph Theory 15 (1991) 223–231. [6] A. Tripathi, S. Vijay, A note on a theorem of Erdös & Gallai, Discrete Math. 265 (2003) 417–420.