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The smallest degree sum that yields graphic sequences with a Z3 -connected realization
European Journal of Combinatorics 34 (2013) 806–811
Contents lists available at SciVerse ScienceDirect
European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc
The smallest degree sum that yields graphic sequences with a Z3 -connected realization✩ Jianhua Yin a , Guodong Guo b a
Department of Mathematics, College of Information Science and Technology, Hainan University, Haikou 570228, PR China
b
Lane Department of Computer Science and Electrical Engineering, West Virginia University, Morgantown, WV 26506, United States
article
abstract
info
A non-increasing sequence π = (d1 , d2 , . . . , dn ) of non-negative integers is said to be graphic if it is the degree sequence of a simple graph G on n vertices. Let A be an (additive) Abelian group. An extremal problem for a graphic sequence to have an A-connected realization is considered as follows: determine the smallest even integer σ (A, n) such that each graphic sequence π = (d1 , d2 , . . . , dn ) with dn ≥ 2 and σ (π ) = d1 + d2 + · · · + dn ≥ σ (A, n) has an A-connected realization. In this paper, we determine σ (Z3 , n) for n ≥ 5. © 2013 Elsevier Ltd. All rights reserved.
Article history: Received 20 June 2012 Accepted 6 January 2013 Available online 31 January 2013
1. Introduction Graphs in this paper are finite, undirected, and loopless. Terms and notation not defined here are from [1]. A graph is simple if it has no multiple edges. Let G be a graph with an orientation D. For a vertex v ∈ V (G), we use E + (v) (or E − (v), respectively) to denote the set of edges with tails (or heads, respectively) at v . Let A be an (additive) Abelian group with identity 0 and A∗ = A − {0}, let F (G, A) denote the set of all functions from E (G) to A, and let F ∗ (G, A) denote the set of all functions from E (G) to A∗ . Given a function f ∈ F (G, A), let ∂ f : V (G) → A be given by
∂ f (v) =
e∈E + (v)
f (e) −
f (e),
e∈E − (v)
where ‘‘ ’’ refers to the addition in A. f ∈ F (G, A) is called an A-flow in G if ∂ f (v) = 0 for each v ∈ V (G). For an edge e ∈ E (G), we call f (e) the flow value of e. The support of f is defined by S (f ) = {e ∈ E (G)|f (e) ̸= 0}. f is called a nowhere-zero
✩ Supported by the National Natural Science Foundation of China (Grant No. 11161016).
E-mail addresses: [email protected], [email protected] (J. Yin), [email protected] (G. Guo). 0195-6698/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2013.01.001
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A-flow if S (f ) = E (G). For an integer k ≥ 2, a nowhere-zero k-flow of G is an integer-valued function f on E (G) such that 0 < |f (e)| < k for each e ∈ E (G), and ∂ f (v) = 0 for each v ∈ V (G). It is well known that G has a nowhere-zero k-flow if and only if it has a nowhere-zero A-flow for |A| = k. Therefore, an A-flow for |A| = k is also called a k-flow. As noted in [3], the existence of a nowhere-zero k-flow of a graph G is independent of the choice of the orientation D. The concept of group connectivity was introduced by Jaeger et al. [4] as a generalization of nowhere-zero flows. A recent survey paper on the topic of group connectivity is [7]. For a graph G, a function b : V (G) → A is called an A-valued zero-sum function on G if v∈V (G) b(v) = 0. The set of all A-valued zero-sum functions on G is denoted by Z (G, A). Given b ∈ Z (G, A) and an orientation D of G, a function f ∈ F ∗ (G, A) is an (A, b)-nowhere-zero flow if ∂ f = b. A graph G is A-connected if G has an orientation D such that, for any b ∈ Z (G, A), G has an (A, b)-nowhere-zero flow. Let ⟨A⟩ be the family of graphs that are A-connected. It is observed in [4] that the property G ∈ ⟨A⟩ is independent of the orientation of G, and that every graph in ⟨A⟩ is 2-edge-connected. In [4], Jaeger et al. made the following conjecture. Conjecture 1.1 ([4]). Every 5-edge connected graph is Z3 -connected. The set of all sequences π = (d1 , d2 , . . . , dn ) of non-negative non-increasing integers with d1 ≤ n − 1 is denoted by NSn . A sequence π ∈ NSn is said to be graphic if it is the degree sequence of a simple graph G on n vertices. In this case, G is said to be a realization of π (or G is said to realize π ). The set of all graphic sequences in NSn is denoted by GSn . If a sequence π consists of the terms d1 , . . . , dt having m m multiplicities m1 , . . . , mt , we may write π = (d1 1 , . . . , dt t ). Let σ (π ) denote the sum of the terms in π . All graphic sequences which have a realization admitting a nowhere-zero 3-flow or 4-flow are characterized in [10,11], respectively. The purpose of this paper is to investigate an extremal problem for π ∈ GSn to have an Aconnected realization. Define σ (A, n) to be the smallest even integer m such that each sequence π = (d1 , d2 , . . . , dn ) ∈ GSn with dn ≥ 2 and σ (π ) ≥ m has an A-connected realization. Recently, Luo et al. [9] obtained the following theorem for |A| = 4. Theorem 1.1 ([9]). Let |A| = 4, n ≥ 3 and π = (d1 , d2 , . . . , dn ) ∈ GSn with dn ≥ 2. If σ (π ) = d1 + d2 + · · · + dn ≥ 3n − 3, then π has an A-connected realization. In fact, Theorem 1.1 establishes the upper bound for σ (A, n) with |A| = 4 as follows:
σ ( A , n) ≤
if n is even, if n is odd.
3n − 2 3n − 3
We can see the lower bound for σ (A, n) with |A| = 4 by taking the graphic sequence
π=
(n − 2, 2n−1 ) (n − 3, 2n−1 )
if n is even, if n is odd,
which has degree sum
σ (π ) =
3n − 4 3n − 5
if n is even, if n is odd.
Let G be a realization of π . It is easy to see that G is not the graph consisting of a bunch of triangles sharing a common vertex and that G contains a vertex whose neighbors are all degree-2 vertices. By Lemma 12 of [9], G is not A-connected. Therefore, we have that
σ (A, n) ≥ σ (π ) + 2 =
3n − 2 3n − 3
Thus we have the following corollary.
if n is even, if n is odd.
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Fig. 1. The graphs that satisfy the Ore-condition and are not Z3 -connected.
Corollary 1.1. If n ≥ 3 and |A| = 4, then
σ (A, n) =
3n − 2 3n − 3
if n is even, if n is odd.
In this paper, we determine σ (Z3 , n) for n ≥ 5. That is, we obtain the following theorem. Theorem 1.2. If n ≥ 5, then
σ ( Z 3 , n) =
5n − 6 5n − 7
if n is even, if n is odd.
2. Proof of Theorem 1.2 In order to prove Theorem 1.2, we need the following known theorems and lemmas. Theorem 2.1 ([8]). If G is a simple graph on n ≥ 3 vertices and G satisfies the Ore-condition that dG (u) + dG (v) ≥ n for each uv ̸∈ E (G), then G is Z3 -connected if and only if G is not one of the 12 graphs shown in Fig. 1. Theorem 2.2 ([10]). Let π = (d1 , d2 , . . . , dn ) be a graphic sequence with d1 ≥ d2 ≥ · · · ≥ dn . If dn ≥ 3 and dn−3 ≥ 4, then π has a Z3 -connected realization. An edge is contracted if it is deleted and its two ends are identified into a single vertex. Let H be a connected subgraph of G. G/H denotes the graph obtained from G by contracting all the edges of H and deleting all the resulting loops. Lemma 2.1 ([6]). Let A be an Abelian group. Then we have the following results. (1) K1 ∈ ⟨A⟩. (2) If H is a connected subgraph of G, and if both H ∈ ⟨A⟩ and G/H ∈ ⟨A⟩, then G ∈ ⟨A⟩. (3) Kn− (n ≥ 5) and Kn (n ≥ 5) are Z3 -connected, where Kn− is the graph obtained from Kn by deleting one edge. (4) A k-cycle Ck is A-connected if and only if |A| ≥ k + 1. Let H1 and H 2 be two subgraphs of a graph G. We say that G is a parallel connection of H1 and H2 , denoted by H1 H2 , if E (H1 ) ∪ E (H2 ) = E (G), |V (H1 ) ∩ V (H2 )| = 2 and |E (H1 ) ∩ E (H2 )| = 1. Lemma 2.2 ([2]). Let G = H1
H2 . If neither H1 nor H2 is Z3 -connected, then G is not Z3 -connected.
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Let π = (d1 , d2 , . . . , dn ) ∈ NSn , and let d′1 ≥ d′2 ≥ · · · ≥ d′n−1 be the rearrangement in nonincreasing order of d1 − 1, d2 − 1, . . . , ddn − 1, ddn +1 , . . . , dn−1 . π ′ = (d′1 , d′2 , . . . , d′n−1 ) is called the residual sequence obtained from π by laying off dn . Lemma 2.3 ([5]). π is graphic if and only if π ′ is graphic. Note that, if dn ≥ 2 and π ′ has an A-connected realization with |A| ≥ 3, then π also has an A-connected realization. Given any two graphs G and H, G ∪ H is the disjoint union of G and H, and G + H, their join, is the graph with V (G + H ) = V (G) ∪ V (H ) and E (G + H ) = E (G) ∪ E (H ) ∪ {uv|u ∈ V (G), v ∈ V (H )}. We now establish the lower bound for σ (Z3 , n) as follows. Lemma 2.4. If n ≥ 5, then
σ (Z3 , n) ≥
5n − 6 5n − 7
if n is even, if n is odd.
Proof. We can establish the lower bound for σ (Z3 , n) by considering the graphic sequence
((n − 1)2 , 3n−2 ) π= ((n − 1)2 , 3n−3 , 2)
if n is even, if n is odd,
which has degree sum
σ (π ) =
5n − 8 5n − 9
if n is even, if n is odd.
This sequence is uniquely realized by
n−2 K2 K2 + 2 G= n−3 K2 + K1 ∪ K2 2
if n is even, if n is odd,
2 2 2 where n− K2 is the disjoint union of n− copies of K2 . In fact, G is the graph of n− copies of K4 meeting 2 2 2
3 copies of K4 and a copy of K3 meeting in a in a common 2 set if n is even, and G is the graph of n− 2 common 2 set if n is odd. Since K4 and K3 are not Z3 -connected, by Lemma 2.2, G is not Z3 -connected. Therefore, we have that
σ (Z3 , n) ≥ σ (π ) + 2 =
5n − 6 5n − 7
if n is even, if n is odd.
Lemma 2.5. Let n ≥ 5 and π = (d1 , d2 , . . . , dn ) ∈ NSn with dn ≥ 3. If σ (π ) = d1 + d2 + · · · + dn = 5n − 7 is even, then π is graphic and has a Z3 -connected realization. Proof. Let π = (d1 , d2 , . . . , dn ) ∈ NSn be a counterexample with n as small as possible. Let s and t denote the number of 4-terms and 3-terms in π respectively. Then dn−(s+t ) ≥ 5. Hence σ (π) = 5n − 7 ≥ 5(n − (s + t )) + 4s + 3t, which implies that s + 2t ≥ 7. Claim 1. n ≥ 7, d1 ≥ 5 and d3 ≥ 4. If n = 5, then σ (π ) = 5 × 5 − 7 = 18. Thus π has a unique realization K5− , so it is graphic and has a Z3 -connected realization. If d1 ≤ 4, then σ (π ) = 5n − 7 ≤ 4n. So n ≤ 7. Thus n = 7 and π = (47 ). Obviously π is graphic, and, by Theorem 2.1, every graph which degree sequence (47 ) is Z3 -connected. Therefore d1 ≥ 5. Now we show that d3 ≥ 4. If d3 ≤ 3, then d3 = 3 and σ (π ) = 5n − 7 ≤ 2(n − 1)+ 3(n − 2) = 5n − 8, a contradiction. If t ≥ 2, let ρ = (d1 − 2, d2 − 1, d3 − 1, d4 , . . . , dn−(s+t ) , 4s , 3t −2 ), and let ρ ∗ = (d∗1 , d∗2 , . . . , d∗n−2 ) be the non-increasing rearrangement of the n − 2 terms of ρ . Then σ (ρ ∗ ) = σ (π )− 10 = 5(n − 2)− 7.
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Since d1 ≥ 5 and d3 ≥ 4, we have d∗n−2 ≥ 3. By the minimality of n, ρ ∗ is graphic and has a realization G which is Z3 -connected. Let u, v, w be three vertices in G with dG (u) = d1 − 2, dG (v) = d2 − 1 and dG (w) = d3 − 1. Let G′ = G ∪ K2 ∪ {x1 u, x1 v, x2 u, x2 w}, where x1 and x2 are the two vertices in K2 . Then G′ is a realization of π . Since G′ /G is a triangle with multiple edges, it is Z3 -connected. Since G is Z3 -connected, by Lemma 2.1(2), G′ is Z3 -connected, a contradiction to the choice of π . If t ≤ 1, then s ≥ 5. If n ≥ 9, let ρ = (d1 − 2, d2 − 1, d3 − 1, d4 , . . . , dn−(s+t ) , 4s−4 , 3t ), and let ρ ∗ = ∗ ∗ (d1 , d2 , . . . , d∗n−4 ) be the non-increasing rearrangement of the n − 4 terms of ρ . Then σ (ρ ∗ ) = σ (π )− 20 = 5(n − 4) − 7. Since n ≥ 9, d1 ≥ 5, and d3 ≥ 4, d∗n−4 ≥ 3. By the minimality of n,, ρ ∗ is graphic and has a Z3 -connected realization G. Let u, v, w be three vertices in G with dG (u) = d1 − 2, dG (v) = d2 − 1, and dG (w) = d3 − 1. Let x1 , x2 , x3 , x4 be the four vertices of a K4 disjoint from G. Let G′ = G ∪ K4 ∪ {x1 u, x2 u, x3 v, x4 w}. Then G′ is a realization of π and G′ /G = K5 . By Lemma 2.1(2), G′ is Z3 -connected, a contradiction. Hence n = 7. Then σ (π ) = 28. Since t ≤ 1, we have π = (47 ) or (5, 45 , 3). It is easy to see that both are graphic and have a Z3 -connected realization, a contradiction again. This contradiction completes the proof of Lemma 2.5. Lemma 2.6. Let n ≥ 5 and π = (d1 , d2 , . . . , dn ) ∈ GSn with dn ≥ 2. If σ (π ) = d1 + d2 + · · · + dn ≥ 5n − 7, then π has a Z3 -connected realization. Proof. Let π = (d1 , d2 , . . . , dn ) be a counterexample with n as small as possible. Claim 1. d1 ≥ d2 ≥ 3, dn = 3 and σ (π ) = 5n − 7. Proof of Claim 1. If d2 < 3, then d2 = 2, and thus 5n − 7 ≤ σ (π ) = d1 + 2(n − 1) ≤ 3(n − 1) = 3n − 3, a contradiction to the hypothesis that n ≥ 5. Hence d1 ≥ d2 ≥ 3. If dn ≥ 4, then, by Theorem 2.2, π has a Z3 -connected realization. So dn ≤ 3. If dn = 2, then π ′ = (d1 − 1, d2 − 1, d3 , . . . , dn−1 ) has degree sum σ (π ′ ) ≥ 5n − 7 − 4 = 5(n − 1)− 6 > 5(n − 1)− 7. Since d1 ≥ d2 ≥ 3, the minimum degree of π ′ is at least 2. By the minimality of n, π ′ has a Z3 -connected realization, and so does π , a contradiction again. This shows that dn = 3. If σ (π ) ≥ 5n − 6, then π ′ = (d1 − 1, d2 − 1, d3 − 1, d4 , . . . , dn−1 ) satisfies σ (π ′ ) ≥ 5n − 6 − 6 = 5(n − 1) − 7. By the minimality of n, π ′ has a Z3 -connected realization, and so does π , a contradiction again. Hence σ (π ) = 5n − 7. This completes the proof of Claim 1. By Claim 1, dn ≥ 3 and σ (π ) = 5n − 7. By Lemma 2.5, π has a Z3 -connected realization, a contradiction to the choice of π . This contradiction completes the proof of Lemma 2.6. Proof of Theorem 1.2. It follows from Lemma 2.6 that
σ (Z3 , n) ≤
5n − 6 5n − 7
if n is even, if n is odd.
By Lemma 2.4, we have
σ (Z3 , n) =
5n − 6 5n − 7
if n is even, if n is odd.
Acknowledgments The authors would like to thank the referees for their helpful suggestions and comments. References [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, American Elsevier, New York, 1976. [2] M. DeVos, R. Xu, G.X. Yu, Nowhere-zero Z3 -flows through Z3 -connectivity, Discrete Math. 306 (2006) 26–30. [3] F. Jaeger, Nowhere-zero flow problems, in: L.W. Beineke, R.J. Wilson (Eds.), Topics in Graph Theory, Vol. 3, Academic Press, London, 1988, pp. 70–95. [4] F. Jaeger, N. Linial, C. Payan, M. Tarsi, Group connectivity of graphs-A nonhomogeneous analogue of nowhere zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165–182. [5] D.J. Kleitman, D.L. Wang, Algorithm for constructing graphs and digraphs with given valences and factors, Discrete Math. 6 (1973) 79–88.
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[6] H.J. Lai, Group connectivity of 3-edge-connected chordal graphs, Graphs Combin. 16 (2000) 165–176. [7] H.J. Lai, X.W. Li, Y.H. Shao, M.Q. Zhan, Group connectivity and group colorings of graphs-A survey, Acta Math. Sin. (Engl. Ser.) 27 (2011) 405–434. [8] R. Luo, R. Xu, J.H. Yin, G.X. Yu, Ore-condition and Z3 -connectivity, European J. Combin. 29 (2008) 1587–1595. [9] R. Luo, R. Xu, G.X. Yu, An extremal problem on group connectivity of graphs, European J. Combin. 33 (2012) 1078–1085. [10] R. Luo, R. Xu, W.A. Zang, C.Q. Zhang, Realizing degree sequences with graphs having nowhere-zero 3-flows, SIAM J. Discrete Math. 22 (2008) 500–519. [11] R. Luo, W.A. Zang, C.Q. Zhang, Nowhere-zero 4-flows, simultaneous edge-colorings, and critical partial latin squares, Combinatorica 24 (2004) 641–657.
Contents lists available at SciVerse ScienceDirect
European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc
The smallest degree sum that yields graphic sequences with a Z3 -connected realization✩ Jianhua Yin a , Guodong Guo b a
Department of Mathematics, College of Information Science and Technology, Hainan University, Haikou 570228, PR China
b
Lane Department of Computer Science and Electrical Engineering, West Virginia University, Morgantown, WV 26506, United States
article
abstract
info
A non-increasing sequence π = (d1 , d2 , . . . , dn ) of non-negative integers is said to be graphic if it is the degree sequence of a simple graph G on n vertices. Let A be an (additive) Abelian group. An extremal problem for a graphic sequence to have an A-connected realization is considered as follows: determine the smallest even integer σ (A, n) such that each graphic sequence π = (d1 , d2 , . . . , dn ) with dn ≥ 2 and σ (π ) = d1 + d2 + · · · + dn ≥ σ (A, n) has an A-connected realization. In this paper, we determine σ (Z3 , n) for n ≥ 5. © 2013 Elsevier Ltd. All rights reserved.
Article history: Received 20 June 2012 Accepted 6 January 2013 Available online 31 January 2013
1. Introduction Graphs in this paper are finite, undirected, and loopless. Terms and notation not defined here are from [1]. A graph is simple if it has no multiple edges. Let G be a graph with an orientation D. For a vertex v ∈ V (G), we use E + (v) (or E − (v), respectively) to denote the set of edges with tails (or heads, respectively) at v . Let A be an (additive) Abelian group with identity 0 and A∗ = A − {0}, let F (G, A) denote the set of all functions from E (G) to A, and let F ∗ (G, A) denote the set of all functions from E (G) to A∗ . Given a function f ∈ F (G, A), let ∂ f : V (G) → A be given by
∂ f (v) =
e∈E + (v)
f (e) −
f (e),
e∈E − (v)
where ‘‘ ’’ refers to the addition in A. f ∈ F (G, A) is called an A-flow in G if ∂ f (v) = 0 for each v ∈ V (G). For an edge e ∈ E (G), we call f (e) the flow value of e. The support of f is defined by S (f ) = {e ∈ E (G)|f (e) ̸= 0}. f is called a nowhere-zero
✩ Supported by the National Natural Science Foundation of China (Grant No. 11161016).
E-mail addresses: [email protected], [email protected] (J. Yin), [email protected] (G. Guo). 0195-6698/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2013.01.001
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A-flow if S (f ) = E (G). For an integer k ≥ 2, a nowhere-zero k-flow of G is an integer-valued function f on E (G) such that 0 < |f (e)| < k for each e ∈ E (G), and ∂ f (v) = 0 for each v ∈ V (G). It is well known that G has a nowhere-zero k-flow if and only if it has a nowhere-zero A-flow for |A| = k. Therefore, an A-flow for |A| = k is also called a k-flow. As noted in [3], the existence of a nowhere-zero k-flow of a graph G is independent of the choice of the orientation D. The concept of group connectivity was introduced by Jaeger et al. [4] as a generalization of nowhere-zero flows. A recent survey paper on the topic of group connectivity is [7]. For a graph G, a function b : V (G) → A is called an A-valued zero-sum function on G if v∈V (G) b(v) = 0. The set of all A-valued zero-sum functions on G is denoted by Z (G, A). Given b ∈ Z (G, A) and an orientation D of G, a function f ∈ F ∗ (G, A) is an (A, b)-nowhere-zero flow if ∂ f = b. A graph G is A-connected if G has an orientation D such that, for any b ∈ Z (G, A), G has an (A, b)-nowhere-zero flow. Let ⟨A⟩ be the family of graphs that are A-connected. It is observed in [4] that the property G ∈ ⟨A⟩ is independent of the orientation of G, and that every graph in ⟨A⟩ is 2-edge-connected. In [4], Jaeger et al. made the following conjecture. Conjecture 1.1 ([4]). Every 5-edge connected graph is Z3 -connected. The set of all sequences π = (d1 , d2 , . . . , dn ) of non-negative non-increasing integers with d1 ≤ n − 1 is denoted by NSn . A sequence π ∈ NSn is said to be graphic if it is the degree sequence of a simple graph G on n vertices. In this case, G is said to be a realization of π (or G is said to realize π ). The set of all graphic sequences in NSn is denoted by GSn . If a sequence π consists of the terms d1 , . . . , dt having m m multiplicities m1 , . . . , mt , we may write π = (d1 1 , . . . , dt t ). Let σ (π ) denote the sum of the terms in π . All graphic sequences which have a realization admitting a nowhere-zero 3-flow or 4-flow are characterized in [10,11], respectively. The purpose of this paper is to investigate an extremal problem for π ∈ GSn to have an Aconnected realization. Define σ (A, n) to be the smallest even integer m such that each sequence π = (d1 , d2 , . . . , dn ) ∈ GSn with dn ≥ 2 and σ (π ) ≥ m has an A-connected realization. Recently, Luo et al. [9] obtained the following theorem for |A| = 4. Theorem 1.1 ([9]). Let |A| = 4, n ≥ 3 and π = (d1 , d2 , . . . , dn ) ∈ GSn with dn ≥ 2. If σ (π ) = d1 + d2 + · · · + dn ≥ 3n − 3, then π has an A-connected realization. In fact, Theorem 1.1 establishes the upper bound for σ (A, n) with |A| = 4 as follows:
σ ( A , n) ≤
if n is even, if n is odd.
3n − 2 3n − 3
We can see the lower bound for σ (A, n) with |A| = 4 by taking the graphic sequence
π=
(n − 2, 2n−1 ) (n − 3, 2n−1 )
if n is even, if n is odd,
which has degree sum
σ (π ) =
3n − 4 3n − 5
if n is even, if n is odd.
Let G be a realization of π . It is easy to see that G is not the graph consisting of a bunch of triangles sharing a common vertex and that G contains a vertex whose neighbors are all degree-2 vertices. By Lemma 12 of [9], G is not A-connected. Therefore, we have that
σ (A, n) ≥ σ (π ) + 2 =
3n − 2 3n − 3
Thus we have the following corollary.
if n is even, if n is odd.
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Fig. 1. The graphs that satisfy the Ore-condition and are not Z3 -connected.
Corollary 1.1. If n ≥ 3 and |A| = 4, then
σ (A, n) =
3n − 2 3n − 3
if n is even, if n is odd.
In this paper, we determine σ (Z3 , n) for n ≥ 5. That is, we obtain the following theorem. Theorem 1.2. If n ≥ 5, then
σ ( Z 3 , n) =
5n − 6 5n − 7
if n is even, if n is odd.
2. Proof of Theorem 1.2 In order to prove Theorem 1.2, we need the following known theorems and lemmas. Theorem 2.1 ([8]). If G is a simple graph on n ≥ 3 vertices and G satisfies the Ore-condition that dG (u) + dG (v) ≥ n for each uv ̸∈ E (G), then G is Z3 -connected if and only if G is not one of the 12 graphs shown in Fig. 1. Theorem 2.2 ([10]). Let π = (d1 , d2 , . . . , dn ) be a graphic sequence with d1 ≥ d2 ≥ · · · ≥ dn . If dn ≥ 3 and dn−3 ≥ 4, then π has a Z3 -connected realization. An edge is contracted if it is deleted and its two ends are identified into a single vertex. Let H be a connected subgraph of G. G/H denotes the graph obtained from G by contracting all the edges of H and deleting all the resulting loops. Lemma 2.1 ([6]). Let A be an Abelian group. Then we have the following results. (1) K1 ∈ ⟨A⟩. (2) If H is a connected subgraph of G, and if both H ∈ ⟨A⟩ and G/H ∈ ⟨A⟩, then G ∈ ⟨A⟩. (3) Kn− (n ≥ 5) and Kn (n ≥ 5) are Z3 -connected, where Kn− is the graph obtained from Kn by deleting one edge. (4) A k-cycle Ck is A-connected if and only if |A| ≥ k + 1. Let H1 and H 2 be two subgraphs of a graph G. We say that G is a parallel connection of H1 and H2 , denoted by H1 H2 , if E (H1 ) ∪ E (H2 ) = E (G), |V (H1 ) ∩ V (H2 )| = 2 and |E (H1 ) ∩ E (H2 )| = 1. Lemma 2.2 ([2]). Let G = H1
H2 . If neither H1 nor H2 is Z3 -connected, then G is not Z3 -connected.
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Let π = (d1 , d2 , . . . , dn ) ∈ NSn , and let d′1 ≥ d′2 ≥ · · · ≥ d′n−1 be the rearrangement in nonincreasing order of d1 − 1, d2 − 1, . . . , ddn − 1, ddn +1 , . . . , dn−1 . π ′ = (d′1 , d′2 , . . . , d′n−1 ) is called the residual sequence obtained from π by laying off dn . Lemma 2.3 ([5]). π is graphic if and only if π ′ is graphic. Note that, if dn ≥ 2 and π ′ has an A-connected realization with |A| ≥ 3, then π also has an A-connected realization. Given any two graphs G and H, G ∪ H is the disjoint union of G and H, and G + H, their join, is the graph with V (G + H ) = V (G) ∪ V (H ) and E (G + H ) = E (G) ∪ E (H ) ∪ {uv|u ∈ V (G), v ∈ V (H )}. We now establish the lower bound for σ (Z3 , n) as follows. Lemma 2.4. If n ≥ 5, then
σ (Z3 , n) ≥
5n − 6 5n − 7
if n is even, if n is odd.
Proof. We can establish the lower bound for σ (Z3 , n) by considering the graphic sequence
((n − 1)2 , 3n−2 ) π= ((n − 1)2 , 3n−3 , 2)
if n is even, if n is odd,
which has degree sum
σ (π ) =
5n − 8 5n − 9
if n is even, if n is odd.
This sequence is uniquely realized by
n−2 K2 K2 + 2 G= n−3 K2 + K1 ∪ K2 2
if n is even, if n is odd,
2 2 2 where n− K2 is the disjoint union of n− copies of K2 . In fact, G is the graph of n− copies of K4 meeting 2 2 2
3 copies of K4 and a copy of K3 meeting in a in a common 2 set if n is even, and G is the graph of n− 2 common 2 set if n is odd. Since K4 and K3 are not Z3 -connected, by Lemma 2.2, G is not Z3 -connected. Therefore, we have that
σ (Z3 , n) ≥ σ (π ) + 2 =
5n − 6 5n − 7
if n is even, if n is odd.
Lemma 2.5. Let n ≥ 5 and π = (d1 , d2 , . . . , dn ) ∈ NSn with dn ≥ 3. If σ (π ) = d1 + d2 + · · · + dn = 5n − 7 is even, then π is graphic and has a Z3 -connected realization. Proof. Let π = (d1 , d2 , . . . , dn ) ∈ NSn be a counterexample with n as small as possible. Let s and t denote the number of 4-terms and 3-terms in π respectively. Then dn−(s+t ) ≥ 5. Hence σ (π) = 5n − 7 ≥ 5(n − (s + t )) + 4s + 3t, which implies that s + 2t ≥ 7. Claim 1. n ≥ 7, d1 ≥ 5 and d3 ≥ 4. If n = 5, then σ (π ) = 5 × 5 − 7 = 18. Thus π has a unique realization K5− , so it is graphic and has a Z3 -connected realization. If d1 ≤ 4, then σ (π ) = 5n − 7 ≤ 4n. So n ≤ 7. Thus n = 7 and π = (47 ). Obviously π is graphic, and, by Theorem 2.1, every graph which degree sequence (47 ) is Z3 -connected. Therefore d1 ≥ 5. Now we show that d3 ≥ 4. If d3 ≤ 3, then d3 = 3 and σ (π ) = 5n − 7 ≤ 2(n − 1)+ 3(n − 2) = 5n − 8, a contradiction. If t ≥ 2, let ρ = (d1 − 2, d2 − 1, d3 − 1, d4 , . . . , dn−(s+t ) , 4s , 3t −2 ), and let ρ ∗ = (d∗1 , d∗2 , . . . , d∗n−2 ) be the non-increasing rearrangement of the n − 2 terms of ρ . Then σ (ρ ∗ ) = σ (π )− 10 = 5(n − 2)− 7.
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Since d1 ≥ 5 and d3 ≥ 4, we have d∗n−2 ≥ 3. By the minimality of n, ρ ∗ is graphic and has a realization G which is Z3 -connected. Let u, v, w be three vertices in G with dG (u) = d1 − 2, dG (v) = d2 − 1 and dG (w) = d3 − 1. Let G′ = G ∪ K2 ∪ {x1 u, x1 v, x2 u, x2 w}, where x1 and x2 are the two vertices in K2 . Then G′ is a realization of π . Since G′ /G is a triangle with multiple edges, it is Z3 -connected. Since G is Z3 -connected, by Lemma 2.1(2), G′ is Z3 -connected, a contradiction to the choice of π . If t ≤ 1, then s ≥ 5. If n ≥ 9, let ρ = (d1 − 2, d2 − 1, d3 − 1, d4 , . . . , dn−(s+t ) , 4s−4 , 3t ), and let ρ ∗ = ∗ ∗ (d1 , d2 , . . . , d∗n−4 ) be the non-increasing rearrangement of the n − 4 terms of ρ . Then σ (ρ ∗ ) = σ (π )− 20 = 5(n − 4) − 7. Since n ≥ 9, d1 ≥ 5, and d3 ≥ 4, d∗n−4 ≥ 3. By the minimality of n,, ρ ∗ is graphic and has a Z3 -connected realization G. Let u, v, w be three vertices in G with dG (u) = d1 − 2, dG (v) = d2 − 1, and dG (w) = d3 − 1. Let x1 , x2 , x3 , x4 be the four vertices of a K4 disjoint from G. Let G′ = G ∪ K4 ∪ {x1 u, x2 u, x3 v, x4 w}. Then G′ is a realization of π and G′ /G = K5 . By Lemma 2.1(2), G′ is Z3 -connected, a contradiction. Hence n = 7. Then σ (π ) = 28. Since t ≤ 1, we have π = (47 ) or (5, 45 , 3). It is easy to see that both are graphic and have a Z3 -connected realization, a contradiction again. This contradiction completes the proof of Lemma 2.5. Lemma 2.6. Let n ≥ 5 and π = (d1 , d2 , . . . , dn ) ∈ GSn with dn ≥ 2. If σ (π ) = d1 + d2 + · · · + dn ≥ 5n − 7, then π has a Z3 -connected realization. Proof. Let π = (d1 , d2 , . . . , dn ) be a counterexample with n as small as possible. Claim 1. d1 ≥ d2 ≥ 3, dn = 3 and σ (π ) = 5n − 7. Proof of Claim 1. If d2 < 3, then d2 = 2, and thus 5n − 7 ≤ σ (π ) = d1 + 2(n − 1) ≤ 3(n − 1) = 3n − 3, a contradiction to the hypothesis that n ≥ 5. Hence d1 ≥ d2 ≥ 3. If dn ≥ 4, then, by Theorem 2.2, π has a Z3 -connected realization. So dn ≤ 3. If dn = 2, then π ′ = (d1 − 1, d2 − 1, d3 , . . . , dn−1 ) has degree sum σ (π ′ ) ≥ 5n − 7 − 4 = 5(n − 1)− 6 > 5(n − 1)− 7. Since d1 ≥ d2 ≥ 3, the minimum degree of π ′ is at least 2. By the minimality of n, π ′ has a Z3 -connected realization, and so does π , a contradiction again. This shows that dn = 3. If σ (π ) ≥ 5n − 6, then π ′ = (d1 − 1, d2 − 1, d3 − 1, d4 , . . . , dn−1 ) satisfies σ (π ′ ) ≥ 5n − 6 − 6 = 5(n − 1) − 7. By the minimality of n, π ′ has a Z3 -connected realization, and so does π , a contradiction again. Hence σ (π ) = 5n − 7. This completes the proof of Claim 1. By Claim 1, dn ≥ 3 and σ (π ) = 5n − 7. By Lemma 2.5, π has a Z3 -connected realization, a contradiction to the choice of π . This contradiction completes the proof of Lemma 2.6. Proof of Theorem 1.2. It follows from Lemma 2.6 that
σ (Z3 , n) ≤
5n − 6 5n − 7
if n is even, if n is odd.
By Lemma 2.4, we have
σ (Z3 , n) =
5n − 6 5n − 7
if n is even, if n is odd.
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