Czechoslovak Mathematical Journal, 59 (134) (2009), 1059–1075
POTENTIALLY Km−G-GRAPHICAL SEQUENCES: A SURVEY
Chunhui Lai,Lili Hu, Fujian (Received May 31, 2008)
Abstract. The set of all non-increasing nonnegative integer sequences π = (d(v1), d(v2), . . . , d(vn)) is denoted by NSn. A sequenceπ∈NSn is said to be graphic if it is the degree sequence of a simple graphGonnvertices, and such a graphGis called a realization ofπ. The set of all graphic sequences in NSn is denoted by GSn. A graphical sequenceπ is potentiallyH-graphical if there is a realization ofπcontainingH as a subgraph, whileπ is forciblyH-graphical if every realization ofπcontainsH as a subgraph. LetKk denote a complete graph onkvertices. LetKm−Hbe the graph obtained fromKmby removing the edges setE(H) of the graph H (H is a subgraph of Km). This paper summarizes briefly some recent results on potentiallyKm−G-graphic sequences and give a useful classification for determiningσ(H, n).
Keywords: graph, degree sequence, potentiallyKm−G-graphic sequences MSC 2000: 05C07, 05C35
1. Introduction
The set of all non-increasing nonnegative integer sequencesπ= (d(v1), d(v2), . . . , d(vn))is denoted byNSn. A sequenceπ∈NSnis said to be graphic if it is the degree sequence of a simple graphGonnvertices, and such a graphGis called a realization ofπ. The set of all graphic sequences inNSnis denoted byGSn. A graphical sequence πis potentiallyH-graphical if there is a realization ofπcontainingH as a subgraph, whileπis forciblyH-graphical if every realization ofπcontainsH as a subgraph. If πhas a realization in which ther+ 1vertices of largest degree induce a clique, then πis said to be potentiallyAr+1-graphic. Letσ(π) =d(v1) +d(v2) +. . .+d(vn),and [x]denote the largest integer less than or equal tox. We denote byG+H the graph Project Supported by NSF of Fujian(2008J0209), Fujian Provincial Training Foundation for “Bai-Quan-Wan Talents Engineering”, Project of Fujian Education Department and Project of Zhangzhou Teachers College (SK07009).
withV(G+H) =V(G)∪V(H)andE(G+H) =E(G)∪E(H)∪ {xy: x∈V(G), y∈ V(H)}. LetKk, Ck, Tk, andPk denote a complete graph onk vertices, a cycle on k vertices, a tree onk+ 1 vertices, and a path onk+ 1 vertices, respectively. Let Fk denote the friendship graph on2k+ 1vertices, that is, the graph ofk triangles intersecting in a single vertex. For06r6t,denote the generalized friendship graph onkt−kr+rvertices byFt,r,k, whereFt,r,k is the graph ofkcopies ofKtmeeting in a commonrset. We use the symbolZ4to denote K4−P2. LetKm−H be the graph obtained fromKm by removing the edges set E(H) of the graphH (H is a subgraph ofKm).
Given a graphH, what is the maximum number of edges of a graph withnvertices not containingH as a subgraph? This number is denotedex(n, H), and is known as the Turán number. In terms of graphic sequences, the number2ex(n, H) + 2is the minimum even integerl such that everyn-term graphical sequenceπwithσ(π)>l is forciblyH-graphical. Erdös, Jacobson and Lehel [13] first consider the following variant: determine the minimum even integer l such that every n-term graphical sequenceπwith σ(π)>l is potentiallyH-graphical. We denote this minimuml by σ(H, n). Erdös, Jacobson and Lehel [13] showed thatσ(Kk, n)>(k−2)(2n−k+1)+2 and conjectured that equality holds. They proved that if π does not contain zero terms, this conjecture is true for k = 3, n > 6. The conjecture was confirmed in [19] and [43]–[46]. Li et al. [46] and Mubayi [55] also independently determined the values σ(Kr,2k) for any k > 3. Li and Yin [51] further determined σ(Kr, n) for r > 7 and n > 2r+ 1. The problem of determining σ(Kr, n) is completely solved.
Gould, Jacobson and Lehel [19] also proved thatσ(pK2, n) = (p−1)(2n−p) + 2 for p > 2; σ(C4, n) = 2[(1/2)(3n−1)] for n > 4. Lai [29] gave a lower bound of σ(Ck, n)and proved that σ(C5, n) = 4n−4 for n > 5 and σ(C6, n) = 4n−2 for n>7. Lai [32] proved that σ(C2m+1, n) =m(2n−m−1) + 2, form>2,n>3m;
and σ(C2m+2, n) = m(2n−m−1) + 4, for m >2, n > 5m−2. Li and Luo [41]
gave a lower bound for σ(3Cl, n) and determined σ(3Cl, n), 4 6l 66, n > l. Li, Luo and Liu [42] determined σ(3Cl, n) for3 6l 68, andn >l. and σ(3C9, n) for n >12. Li and Yin [48] determined σ(3Cl, n) forn sufficiently large. Yin, Li and Chen [68] determined σ(kCl, n), l > 7, 3 6 k 6 l. Chen and Yin [9] determined the valuesσ(W5, n)forn>11,whereWr is a wheel graph on rvertices. Forr×s complete bipartite graphKr,s, Gould, Jacobson and Lehel [19] determinedσ(K2,2, n).
Yin et al. [63], [65], [69], [70] determined σ(Kr,s, n) for s >r > 2 and sufficiently largen. For r×s×t complete 3-partite graph Kr,s,t, Erdös, Jacobson and Lehel [13] determinedσ(K1,1,1, n). Lai [30] determinedσ(K1,1,2, n). Yin [58] and Lai [34]
independently determinedσ(K1,1,3, n). Chen [7] determinedσ(K1,1,t, n) fort >3, n>2[14(t+ 5)2] + 3. Chen[5] determinedσ(K1,2,2, n)for56n68andσ(K2,2,2, n)
for n >6. LetKst denote the complete t partite graph such that each partite set has exactlys vertices. Guantao Chen, Michael Ferrara, Ronald J. Gould and John R. Schmitt [11] showed thatσ(Kst, n) =π(K(t−2)s+Ks,s, n)and obtained the exact value ofσ(Kj+Ks,s, n)fornsufficiently large. Consequently, they obtained the exact value ofσ(Kst, n)fornsufficiently large. Forn>5, Ferrara, Jacobson and Schmitt [17] determined σ(Fk, n)where Fk denotes the graph of k triangles intersecting at exactly one common vertex. In [16], Ferrara, Gould and Schmitt determined a lower bound forσ(Kst, n),whereKstdenotes the complete multipartite graph withtpartite sets each of sizes, and proved equality in the cases= 2. They also provided a graph theoretic proof for the value ofσ(Kt, n). Michael J. Ferrara [15] determinedσ(H, n) for the graph H = Km1 ∪Km2 ∪. . .∪Kmk, where n is sufficiently large integer.
Ferrara, M., Jacobson, M., Schmitt, J. and Siggers M. [18] determinedσ(Ks,t, m, n), σ(Pt, m, n) and σ(C2t, m, n) where σ(H, m, n) is the minimum integer k such that every bigraphic pairS= (A, B)with |A|=m, |B|=nandσ(S)>kis potentially H-bigraphic. For an arbitrarily chosen H, Schmitt, J. R. and Ferrara, M. [56] gave a good lower bound forσ(H, n). Yin and Li [67] determinedσ(Kr1,r2,...,rl,r,s,n)for sufficiently largen. Moreover, Yin, Chen and Schmitt [62] determined σ(Ft,r,k, n) fork>2, t>3,16r6t−2 and sufficiently large, whereFt,r,k denotes the graph of kcopies of Ktmeeting in a common r set. Gupta, Joshi and Tripathi [20] gave a necessary and sufficient condition for the existence of a tree of order n with a given degree set. Yin [59] gave a new necessary and sufficient condition for π to be potentiallyKr+1-graphic. Jiong-sheng Li and Jianhua Yin [50] gave a survey on graphical sequences.
2. Potentially Km−G-graphical sequences
LetH be a graph withmvertices, thenH =Km−(Km−H).LetG=Km−H, then σ(H, n) = σ(Km−G, n). If Problems 1–5 in the Open Problems section are solved, then the problem of determining σ(H, n) is completely solved. We think Problems 3 and 4 are a useful classification for determiningσ(H, n).
Gould, Jacobson and Lehel [19] pointed out that it would be nice to see where in the range from3n−2 to 4n−4 the value σ(K4−e, n) lies. Later, Lai [30] proved that
Theorem 1. Forn= 4,5andn>7
σ(K4−e, n) =
(3n−1 ifnis odd, 3n−2 ifnis even.
Forn= 6,ifSis a 6-term graphical sequence withσ(S)>16, then either there is a realization ofS containingK4−eor S= (36). (Thusσ(K4−e,6) = 20.)
Huang [26] gave a lower bound ofσ(Km−e, n). Yin, Li and Mao [71] and Huang [27] independently determined the valuesσ(K5−e, n)as follows.
Theorem 2. Ifn>5, then
σ(K5−e, n) =
(5n−7, ifnis odd, 5n−6, ifnis even.
Lai [35]–[36] determinedσ(K5−C4, n), σ(K5−P3, n)andσ(K5−P4, n).
Theorem 3. Forn>5,σ(K5−C4, n) =σ(K5−P3, n) =σ(K5−P4, n) = 4n−4.
Yin and Li [66] gave a good method for determining the values σ(Kr+1−e, n) (in fact, Yin and Li [66] also determined the valuesσ(Kr+1−ke, n)for r>2 and n>3r2−r−1).
Theorem 4. Let n >r+ 1 and π = (d1, d2, . . . , dn) ∈ GSn with dr+1 > r. If di>2r−ifori= 1,2, . . . , r−1, then πis potentiallyAr+1-graphic.
Theorem 5. Let n>2r+ 2 and π= (d1, d2, . . . , dn)∈GSn with dr+1 >r. If d2r+2>r−1, thenπis potentially Ar+1-graphic.
Theorem 6. Letn>r+ 1andπ= (d1, d2, . . . , dn)∈GSn with dr+1>r−1. If di>2r−ifori= 1,2, . . . , r−1, then πis potentiallyKr+1−e-graphic.
Theorem 7. Letn >2r+ 2 and π= (d1, d2, . . . , dn)∈GSn with dr−1 >r. If d2r+2>r−1, thenπis potentially Kr+1−e-graphic.
Theorem 8. Ifr>2andn>3r2−r−1,then
σ(Kr+1−ke, n) =
((r−1)(2n−r)−(n−r) + 1 ifn−r is odd, (r−1)(2n−r)−(n−r) + 2 ifn−r is even.
After reading [66], Yin [72] determined the values σ(Kr+1−K3, n) for r > 3, n>3r+ 5.
Theorem 9. Ifr>3andn>3r+ 5,thenσ(Kr+1−K3, n) = (r−1)(2n−r)− 2(n−r) + 2.
Determiningσ(Kr+1−H, n), whereH is a tree on 4 vertices, is more useful than a cycle on 4 vertices (for example,C46⊂Ci, butP3⊂Cifori>5). So, after reading [66] and [72], Lai and Hu [38] determinedσ(Kr+1−H, n) for n>4r+ 10, r >3, r+ 1>k>4andH a graph onkvertices which containing a tree on4 vertices but does not contain a cycle on3 vertices andσ(Kr+1−P2, n)forn>4r+ 8,r>3.
Theorem 10. Ifr>3andn>4r+ 8, thenσ(Kr+1−P2, n) = (r−1)(2n−r)− 2(n−r) + 2.
Theorem 11. Ifr>3,r+ 1>k>4and n>4r+ 10, then σ(Kr+1−H, n) = (r−1)(2n−r)−2(n−r),whereH is a graph onkvertices which contains a tree on 4vertices but not contains a cycle on3 vertices.
There are a number of graphs onkvertices which contain a tree on4 vertices but do not containing a cycle on3vertices (for example, the cycle onkvertices, the tree onkvertices, and the complete 2-partite graph onkvertices, etc).
Lai and Sun [39] determinedσ(Kr+1−(kP2∪tK2), n) forn>4r+ 10,r+ 1>
3k+ 2t,k+t>2,k>1,t>0.
Theorem 12. If n >4r+ 10, r+ 1 > 3k+ 2t, k+t >2, k > 1, t > 0, then σ(Kr+1−(kP2∪tK2), n) = (r−1)(2n−r)−2(n−r).
As yet, the problem of determiningσ(Kr+1−H, n)forH not containing a cycle on 3 vertices andnsufficiently large has not been solved.
Lai [37] determinedσ(Kr+1−Z, n)for n>5r+ 19,r+ 1>k>5,j >5and Z a graph onk vertices and j edges which contains a graphZ4 but does not contain a cycle on 4 vertices. In the same paper, the author also determined the values of σ(Kr+1−Z4, n),σ(Kr+1−(K4−e), n)andσ(Kr+1−K4, n)forn>5r+ 16,r>4.
Theorem 13. Ifr>4 andn>5r+ 16, then
σ(Kr+1−K4, n) =σ(Kr+1−(K4−e), n) =
σ(Kr+1−Z4, n) =
((r−1)(2n−r)−3(n−r) + 1 ifn−ris odd, (r−1)(2n−r)−3(n−r) + 2 ifn−ris even.
Theorem 14. Ifn>5r+ 19,r+ 1>k>5,andj>5, then σ(Kr+1−Z, n) =
((r−1)(2n−r)−3(n−r)−1 ifn−ris odd, (r−1)(2n−r)−3(n−r)−2 ifn−ris even
where Z is a graph on k vertices and j edges which contains a graphZ4 but does not contain a cycle on4 vertices.
There are a number of graphs on k vertices and j edges which contain a graph Z4 but do not contain a cycle on 4 vertices. (For example, the graph obtained by C3, Ci1, Ci2, . . . , Cip intersecting in a single vertex (ij 6= 4, j = 1,2,3, . . . , p)(if ij = 3, j = 1,2,3, . . . , p, then this graph is the friendship graphFp+1), the graph obtained by C3, Pi1, Pi2, . . . , Pip intersecting in a single vertex (i1 >1), the graph obtained by C3, Pi1, Ci2, . . . , Cip (ij 6= 4, j = 2,3, . . . , p, i1 > 1) intersecting in a single vertex, etc.)
Lai and Yan [40] proved that
Theorem 15. Ifn>5r+ 18,r+ 1>k>7,andj>6, then σ(Kr+1−U, n) =
((r−1)(2n−r)−3(n−r)−1 ifn−ris odd, (r−1)(2n−r)−3(n−r) ifn−ris even
whereU is a graph onkvertices andj edges which contains a graph(K3∪P3)but does not contain a cycle on4 vertices and not containsZ4.
There are a number of graphs on k vertices andj edges which contains a graph (K3∪P3) but do not contain a cycle on 4 vertices and do not contain Z4. (For example,C3∪Ci1∪Ci2∪. . .∪Cip(ij6= 4, j= 2,3, . . . , p, i1>5),C3∪Pi1∪Pi2∪. . .∪Pip
(i1>3),C3∪Pi1∪Ci2∪. . .∪Cip (ij6= 4,j= 2,3, . . . , p,i1>3), etc.)
A harder question is to characterize the potentiallyH-graphic sequences without zero terms. Luo [53] characterized the potentially Ck-graphic sequences for each k= 3,4,5.
Theorem 16. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>3. Then π is potentially C3-graphic if and only if d3 > 2 except for 2 case: π = (24) and π= (25).
Theorem 17. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>4. Then πis potentiallyC4-graphic if and only if the following conditions hold:
(1) d4>2.
(2) d1=n−1impliesd2>3.
(3) Ifn= 5,6,thenπ6= (2n).
Theorem 18. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentiallyC5-graphic if and only if the following conditions hold:
(1) d5>2.
(2) Fori= 1,2,d1=n−iimpliesd4−i>3.
(3) Ifπ= (d1, d2,2k,1n−k−2),thend1+d26n+k−2.
Chen [2] characterized the potentiallyCk-graphic sequences fork= 6.
Theorem 19. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>6. Then πis potentiallyC6-graphic if and only if the following conditions hold:
(1) d6>2.
(2) Ifn= 7,8,thenπ6= (2n).
(3) Fori= 1,2,3,d1=n−i impliesd5−i>3.
(4) Ifπ= (d1, d2,2k,1n−k−2),thend1+d26n+k−2; ifπ= (d1, d2,3,2k,1n−k−3), thend1+d26n+k; ifπ= (d1, d2,3,3,2k,1n−k−4),thend1+d26n+k+ 2.
Yin, Chen and Chen [60] characterized the potentiallykCl-graphic sequences for eachk= 3,46l65and k= 4,l= 5.
Theorem 20. Let π = (d1, d2, . . . , dn) ∈ GSn be a potentially C4-graphic se- quence. Thenπis potentially3C4-graphic if and only ifπsatisfies one of the following conditions:
(1) d2>3 andπ6= (32,24);
(2) π= (d1,2k,1n−k−1)with26d163andk>6,andπ6= (28)and(29);
(3) π= (d1,2k,1n−k−1)with46d16n−2andk>5,andπ6= (4,26)and(4,27).
Theorem 21. Let π = (d1, d2, . . . , dn) ∈ GSn be a potentially C5-graphic se- quence. Thenπis potentially3C5-graphic if and only ifπsatisfies one of the following conditions:
(1) d2>3 andπ6= (32,24)and (32,25);
(2) π= (d1,2k,1n−k−1)with26d163andk>11,andπ6= (213)and(214);
(3) π= (d1,2k,1n−k−1)with46d165andk>10,andπ6= (4,211)and (4,212);
(4) π = (d1,2k,1n−k−1) with 6 6 d1 6 n−4 and k > 9, and π 6= (4,210) and (4,211).
Theorem 22. Let π = (d1, d2, . . . , dn) ∈ GSn be a potentially C5-graphic se- quence. Thenπis potentially4C5-graphic if and only ifπsatisfies one of the following conditions:
(1) d2>3;
(2) π= (d1,2k,1n−k−1)with26d163andk>8,andπ6= (210)and(211);
(3) π= (d1,2k,1n−k−1)with46d16n−4andk>7,andπ6= (4,28)and(4,29).
Chen, Yin and Fan [10] characterized the potentially kCl-graphic sequences for each36k65, l= 6.
Theorem 23. Let π = (d1, d2, . . . , dn) ∈GSn, n >6, and π 6= (32,210), (219), (220),(4,217), (4,218), (6,216),(6,217), (8,215),(8,216).Then πis potentially3C6- graphic if and only ifπ be a potentiallyC6-graphic sequence, andπsatisfies one of the following conditions:
(1) d3>3, and ifd1=d3= 3,d4= 2,thend10= 2;
(2) d2>4,d3= 2,d7= 2;
(3) d2 = 3, d3 = 2, and if4 >d1 >3,then d10 = 2, and if n−4 >d1 >5, then d9= 2;
(4) d2= 2,and if3>d1>2,thend18= 2,and if5>d1>4,thend17= 2, and if 7>d1>6,thend16= 2,and ifn−7>d1>8,thend15= 2.
Theorem 24. Let π = (d1, d2, . . . , dn) ∈ GSn, n > 6, and π 6= (216), (217), (4,214),(4,215), (6,213), (6,214), Thenπ is potentially4C6-graphic if and only if π is a potentiallyC6-graphic sequence, andπsatisfies one of the following conditions:
(1) d3>3, and ifd1=d3= 3,d4= 2,thend10= 2;
(2) d2>4,d3= 2,d7= 2;
(3) d2 = 3, d3 = 2, and if4 >d1 >3,then d10 = 2, and if n−4 >d1 >5, then d9= 2;
(4) d2= 2,and if3>d1>2,thend15= 2,and if5>d1>4,thend14= 2, and if n−7>d1>6,thend13= 2.
Theorem 25. Let π = (d1, d2, . . . , dn) ∈ GSn, n > 6, and π 6= (212), (213), (4,210), (4,211), Then πis potentially 5C6-graphic if and only if π is a potentially C6-graphic sequence, and πsatisfies one of the following conditions:
(1) d2>3;
(2) 3>d1>2,d2= 2,d11= 2;
(3) n−6>d1>4,d2= 2,d10= 2.
Luo and Warner [54] characterized the potentiallyK4-graphic sequences.
Theorem 26. Letπ= (d1, d2, . . . , dn)be a graphic sequence without zero terms and with d4 >3 andn>4. Thenπis potentiallyK4-graphic if and only ifd4>3 andπ6= (n−1,3s,1n−s−1)for eachs= 4,5except the following sequences:
n= 5 : (4,34), (34,2);
n= 6: (46),(42,34),(4,34,2),(36),(35,1),(34,22);
n= 7: (47),(43,34),(4,36),(4,35,1),(36,2),35,2,1);
n= 8: (37,1),(36,12).
Eschen and Niu [14] and Lai [31] independently characterized the potentiallyK4− e-graphic sequences.
Theorem 27. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>4. Then πis potentially(K4−e)-graphic if and only if the following conditions hold:
(1) d2>3.
(2) d4>2.
(3) Ifn= 5,6,thenπ6= (32,2n−2)andπ6= (36).
Yin and Yin [73] characterized the potentiallyK5−eandK6-graphic sequences.
Theorem 28. Let n >5 and π = (d1, d2, . . . , dn) ∈NSn be a positive graphic sequence with d3 > 4 and d5 > 3. Then π is potentially K5−e-graphic if and only ifπis not one of the following sequences: (n−1,46,1n−7),(n−1,42,34,1n−7), (n−1,42,33,1n−6);
n= 6: (46),(44,32),(43,32,2);
n= 7: (43,34), (52,4,34), (47),(45,32), (5,43,33), (52,45),(5,45,3),(43,32,22), (44,32,2),(5,42,33,2),(46,2),(43,33,1);
n= 8: (58),(48),(52,46), (6,47), (44,34),(5,42,35),(46,32),(5,46,3),(43,34,2), (47,2),(44,33,1),(5,42,34,1),(43,33,2,1),(46,3,1),(5,46,1);
n= 9: (49),(43,35,1),(48,2),(47,3,1),(5,47,1),(43,34,12),(47,12);
n= 10: (48,12).
Theorem 29. Let n > 18 and π = (d1, d2, . . . , dn) ∈ NSn be a positive graphic sequence with d6 > 5. Then π is potentially A6-graphic if and only if π66∈ {(2),(22),(3,1),(32),(3,2,1),(32,2),(33,1),(32,12)}.
Yin and Chen [61] characterized the potentiallyKr,s-graphic sequences forr= 2, s= 3andr= 2,s= 4.
Theorem 30. Letn>5 andπ= (d1, d2, . . . , dn)∈GSn. Then πis potentially K2,3-graphic if and only ifπsatisfies the following conditions:
(1) d2>3 andd5>2;
(2) ifd1=n−1 andd2= 3, thend5= 3;
(3) π6= (32,24), (32,25),(43,23),(n−1,35,1n−6)and(n−1,36,1n−7).
Theorem 31. Letn>6 andπ= (d1, d2, . . . , dn)∈GSn. Then πis potentially K2,4-graphic if and only ifπsatisfies the following conditions:
(1) d2>4 andd6>2;
(2) ifd1=n−1 andd2= 4, thend3= 4andd6>3;
(3) π 6= (43,24), (42,25), (42,26), (52,4,24), (53,3,23), (6,52,25), (53,24,1), (63,26), (n− 1,42,34,1n−7), (n −1,42,35,1n−8), (n −2,42,23,1n−6), and (n−2,43,22,1n−6).
Chen [3] characterized the potentiallyK5−2K2-graphic sequences for56n68.
Hu and Lai [23] characterized the potentiallyK5−P3,K5−A3,K5−K3,K5−K1,3
andK5−2K2-graphic sequences whereA3 isP2∪K2.
Theorem 32. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentiallyK5−P3-graphic if and only if the following conditions hold:
(1) d1>4,d3>3 andd5>2.
(2) π6= (4,32,23),(4,32,24)and(4,36).
Theorem 33. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentiallyK5−A3-graphic if and only if the following conditions hold:
(1) d4>3 andd5>2.
(2) π6= (n−1,33,2n−k,1k−4)where n>6 and k= 4,5, . . . , n−2, n and khave the same parity.
(3) π 6= (34,22), (36), (34,23), (36,2), (4,36), (37,1), (38), (n−1,35,1n−6) and (n−1,36,1n−7).
Theorem 34. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentiallyK5−K3-graphic if and only if the following conditions hold:
(1) d2>4 andd5>2.
(2) π6= (42,24), (42,25),(43,23)and(46).
Theorem 35. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentiallyK5−K1,3-graphic if and only if the following conditions hold:
(1) d1>4 andd4>3.
(2) π6= (4,34,2), (46), (42,34),(4,36), (47), (4,35,1),(n−1,34,1n−5)and(n−1, 35,1n−6).
Theorem 36. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentiallyK5−2K2-graphic if and only if the following conditions hold:
(1) d1>4 andd5>3;
(2)
π6=
((n−i, n−j,3n−i−j−2k,22k,1i+j−2), n−i−j is even;
(n−i, n−j,3n−i−j−2k−1,22k+1,1i+j−2), n−i−j is odd.
where16j6n−5and06k6[12(n−j−i−4)].
(3) π 6= (42,34), (4,34,2), (5,4,35), (5,35,2), (47), (43,34), (42,34,2), (4,36), (4,35,1),(4,34,22), (5,37), (5,36,1), (48), (42,36), (42,35,1), (4,36,2), (4,35, 2,1),(4,37,1),(4,36,12), (n−1,35,1n−6)and(n−1,36,1n−7).
Hu and Lai [21] characterized the potentiallyK5−C4-graphic sequences.
Theorem 37. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentially(K5−C4)-graphic if and only if the following conditions hold:
(1) d1>4.
(2) d5>2.
(3) π6= ((n−2)2,2n−2)for n>6, where the symbolxy stands for y consecutive termsx.
(4) π 6= (n−k, k+i,2i,1n−i−2) where i = 3,4, . . . , n−2k and k = 1,2, . . . , [12(n−1)]−1.
(5) Ifn= 6, then π6= (4,25).
(6) Ifn= 7, then π6= (4,26).
Hu and Lai [22] characterized the potentiallyK5−Z4-graphic sequences.
Theorem 38. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentially(K5−Z4)-graphic if and only if the following conditions hold:
(1) d1>4,d2>3 andd4>2.
Hu, Lai and Wang [25] characterized the potentiallyK5−P4andK5−Y4-graphic sequences whereY4is a tree on 5 vertices and 3 leaves.
Theorem 39. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentiallyK5−P4-graphic if and only if the following conditions hold:
(1) d2>3.
(2) d5>2.
(3) π6= (n−1, k,2t,1n−2−t)where n>5, k, t= 3,4, . . . , n−2, and,kand thave different parities.
(4) For n > 5, π 6= (n−k, k+i,2i,1n−i−2) where i = 3,4, . . . , n−2k and k = 1,2, . . . ,[12(n−1)]−1.
(5) Ifn= 6,7, thenπ6= (32,2n−2).
Theorem 40. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentiallyK5−Y4-graphic if and only if the following conditions hold:
(1) d3>3.
(2) d4>2.
(3) π6= (36).
Hu and Lai [24] characterized the potentiallyK3,3andK6−C6-graphic sequences.
Theorem 41. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>6. Then πis potentiallyK3,3-graphic if and only if the following conditions hold:
(1) d6>3;
(2) fori= 1,2,d1=n−iimpliesd4−i>4;
(3) d2=n−1impliesd3>5 ord6>4;
(4) d1+d2= 2n−ianddn−i+3= 1(36i6n−4) impliesd3>5or d6>4;
(5) d1+d2= 2n−ianddn−i+4= 1(46i6n−3) impliesd3>4;
(6) π= (d1, d2,34,2t,1n−6−t)or(d1, d2,42,32,2t,1n−6−t)impliesd1+d26n+t+2;
(7) π= (d1, d2,4,34,2t,1n−7−t)impliesd1+d26n+t+ 3;
(8) fort = 5,6,π6= (n−i, k+i,4t,2k−t,1n−2−k)where i= 1, . . . ,[12(n−k)] and k=t, . . . , n−2i;
(9) π 6= (54,32,2), (46), (36,2), (64,34), (42,36), (4,36,2), (36,22), (38), (37,1), (4,38), (4,37,1), (38,2),(37,2,1), (39,1), (38,12), (n−1,42,34,1n−7),(n−1, 42,35,1n−8),(n−1,53,33,1n−7),(n−2,4,35,1n−7),(n−2,4,36,1n−8),(n−3, 36,1n−7), (n−3,37,1n−8).
Theorem 42. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>6. Then πis potentiallyK6−C6-graphic if and only if the following conditions hold:
(1) d6>3;
(2) fori= 1,2,d1=n−iimpliesd4−i>4;
(3) d2=n−1impliesd4>4;
(4) d1+d2= 2n−ianddn−i+3= 1(36i6n−4) impliesd4>4;
(5) d1+d2= 2n−ianddn−i+4= 1(46i6n−3) impliesd3>4;
(6) π= (d1, d2, d3,3k,2t,1n−3−k−t)impliesd1+d2+d36n+ 2k+t+ 1;
(7) π= (d1, d2,34,2t,1n−6−t)impliesd1+d26n+t+ 2;
(8) π 6= (n−i, k, t,3t,2k−i−t−1,1n−2−k+i) where i = 1, . . . ,[12(n−t−1)] and k=i+t+ 1, . . . , n−iandt= 4,5, . . . , k−i−1;
(9) π 6= (36,2), (42,36), (4,36,2), (36,22), (38), (37,1), (4,38), (4,37,1), (38,2), (37,2,1), (39,1), (38,12), (n−1,42,34,1n−7), (n−1,42,35,1n−8), (n−2, 4,35,1n−7),(n−2,4,36,1n−8), (n−3,36,1n−7), (n−3,37,1n−8).
Xu and Hu [57] characterized the potentially K1,4+e-graphic sequences. Chen and Li [8] characterized the potentiallyK1,t+e-graphic sequences.
Theorem 43. Letπ= (d1, d2, . . . , dn)be a graphic sequence withn>5. Then πis potentiallyK1,4+e-graphic if and only if d1>4,d3>2.
Theorem 44. Lett>3,π= (d1, d2, . . . , dn)is a graphic sequence withn>t+ 1.
Thenπis potentiallyK1,t+e-graphic if and only ifd1>t,d3>2.
Open problems
Problem 1. Determineσ(Kr+1−G, n)forGis a graph onkvertices andjedges which contains a graphK3∪K1,3but does not contain a cycle on4vertices and does not containZ4 andP3.
Problem 2. Determineσ(Kr+1−G, n)forG=K3∪iK2∪jP2∪tK3.
Problem 3. Determineσ(Kr+1−G, n)for graphGwhich containsC3,C4, . . . , Cl
but does not contain a cycle onl+ 1vertices (46l6r).
Problem 4. Determine σ(Kr+1 −G, n) for a graph G which contains C3, C4, . . . , Cr+1.
Problem 5. Determineσ(Kr+1−G, n)for smalln.
Problem 6. Characterize potentiallyKr+1−G-graphic sequences for the remain- ingG.
Acknowledgment. The first author is particularly indebted to Professor Jiong- sheng Li for introducing him to degree sequences. The authors wish to thank Pro- fessor Gang Chen, R. J. Gould, Jiongsheng Li, Rong Luo, John R. Schmitt, Zi-Xia Song, Amitabha Tripathi, Jianhua Yin and Mengxiao Yin for sending some of their papers to us.
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Authors’ addresses: C h u n h u i L a i (correspondent author), Department of Math- ematics, Zhangzhou Teachers College, Zhangzhou, Fujian 363000, P. R. China, e-mail:
[email protected]; L i l i H u, Department of Mathematics, Zhangzhou Teach- ers College, Zhangzhou, Fujian 363000, P. R. China, e-mail:[email protected].
