aa r X i v : . [ m a t h . C O ] O c t Edge number report 1: state of the artestimates for n . J¨orgen BackelinApril 22, 2019
Abstract
This first extracted report contains all lower and upper bounds for e-numberse(3 , k ; n ), for n
43, that I know. All but 24 of them are known (exactly). Verylittle of the proofs is given. A few consequences for upper classical Ramsey numberbounds are mentioned.
Throughout the years, I have investigated e-numbers, and updated my tables of theseand of properties for graphs with edge numbers close to the respective e-number. Theresults have been collected in the various updated versions of [1]. However, that work isnot easily accessible; not only since I have not made it public, but since it is large, andbased on a somewhat complex terminology, both for graph objects and for methods fordealing with them.At present, I’m integrating the consequences of Goedgebeur’s and Radziszowski’s in-vestigations in [4] into my tables. This is slow work; I have now more or less finished itup to vertex number 43. This has yielded a few improvements, compared both to [4] andto older versions of [1].I have received some criticism for not making my results more accessible. In thisreport, I indeed try to present the more recent ones, as regards e-number bounds; but notthe further Ramsey graph properties. I believe that this makes it easier to uaccess theconclusions ; but it makes it harder to reproduce or improve the proofs . I outline a fewproof examples; they may at least illustrate the ‘Ramsey calculus’ methods.Moreover I also discuss upper bounds for e-numbers. This is an area not equally wellcovered by the literature, I think, and I’m not sure of how good the upper bounds I givehere are, compared to the state-of-the-art.Finally, the terminology is a bit experimentative. I try to make it more conformant toother recent state-of-the-art articles, and (against my instincts) leave a good bit undefined.I’ll be very thankful for comments, both on this, and on the factual content of this report.1
Definitions.
Throughout this work, all graphs G = ( V, E ) are finite, simple, and undirected; and theyare triangle-free ; i. e., the clique number ω ( G ) second degree of a vertex v in a graph G isdeg ( v ) = deg G ( v ) := X w ∈ N ( v ) deg( w ) , where N ( v ) is the set of vertices adjacent to v . (The second degree is denoted Z ( v ) ine. g. [4].) The induced G subgraph on V \ ( N ( v ) ∪ { v } ) is denoted G v . G is an ( i, j ; n, e ) -graph and an ( i, j ; n ) -graph , if ω ( G ) < i , its independence number α ( G ) < j , n ( G ) := | V | = n , and e ( G ) := | E | = e .For any positive integers i , j , and n , the e-number e( i, j ; n ) is the minimal number e ,such that there are ( i, j ; n, e )-graphs, or ∞ , if no ( i, j ; n )-graphs exist. They are of greatinterest for finding improved bounds of Ramsey numbersR( i, j ) := min( n : e( i, j ; n ) = ∞ ) , but are also of interest in themselves.In this report, we only discuss the e-numbers e(3 , j ; n ). For the estimates, we shall usea few linear or ‘piecewise linear’ functions on two integer variables, namely, f ( n, k ) = max (0 , n − k, n − k, n − k, n − k ) ; f ( n, k ) = 8 n − . k ; f ( n, k ) = 9 n − k ; and f ( n, k ) = 6 . n − . k . Note, that f ( n, k ) = 6 n − k , if n > k .Occasionally, we mention the “linear graph invariant” t ( G ) := e ( G ) − n ( G ) + 13 α ( G ) . W , denotes the cyclic graph with 13 vertices (conventionally named u , . . . , u ),and with two vertices forming an edge if the absolute value of their indices counted modulo13 is either 1 or 5. (This graph very often is denoted H .)For other concepts, background, et cetera, see the bibliography. In particular, weshall discuss some graphs given by means of extension patterns , which provide recipiesfor constructing them step-by-step; but neither the patterns and nor the correspondinggraphs are formally described here. 2 Known general values.
For n . k + 1 .
5, all e-numbers are known. (This indeed includes all e(3 , k + 1; n ) with n
43 and k > Proposition 1.
For all positive integers n and k , e(3 , k + 1; n ) > f ( n, k ) . The values are exact if and only if n <
R(3 , k + 1) , and moreover either n . k − ,or n = 3 . n . For a proof, see e. g. [10]. Note, that part of the result is the fact that t ( G ) > G . Lemma 3.1.
Let k and n be positive integers, such that k n < R(3 , k + 1) , but e(3 , k + 1; n ) > f ( n, k ) . Then e(3 , k + 1; n ) = f ( n, k ) + 1 ⇐⇒ − < n − . k < , e(3 , k + 1; n ) = f ( n, k ) + 2 ⇐⇒ < n − . k . , and e(3 , k + 1; n ) > f ( n, k ) +3 ⇐⇒ . < n − . k , The proof depends on deriving properties for graphs with t ( G )
2. In [1], indeed,all G with t ( G ) t ( G ) = 2. (Actually, the complete characterising of the graphs with t ( G ) = 0 also is themain object of the stand-alone manuscript [2]. The t ( G ) = 2 result partly employs [4].)Employing some constructions, we find that the lower bound in the last part oflemma 3.1 is exact in a few cases: Lemma 3.2. If k n < R(3 , k + 1) and . < n − . k . , then e(3 , k + 1; n ) = f ( n, k ) + 3 . If n > . k + 1 .
5, and moreover k
12, then e(3 , k + 1; n ) > f ( n, k ) + 3; and I findit likely that this should hold also for all higher k . Moreover, I guess thate(3 , k + 1; n ) > max( f ( n, k ) , f ( n, k )) , (1)too; but I am far from being able to prove this. The best general result I have for n − . k ≫ Lemma 3.3.
For any n and k , e(3 , k + 1; n ) > f ( n, k ) . (This is contained in [1, proposition 13.5], which is proved by means of a somewhatcomplicated induction argument). 3 The other values for n . For n
34, all e(3 , k + 1; n ) are known. Actually, only 15 of them are ‘sporadic’, i. e.,not given by the known Ramsey numbers, or in section 3; and they all have n >
22 and6 k
9. Thus, they are included in the following e(3 , l ; n ) table (where l = k + 1): n \ l ∞
49 35 2524 ∞
56 40 3025 ∞
65 46 3526 ∞
73 52 4027 ∞
85 61 4528 ∞ ∞
68 5129 ∞ ∞
77 5830 ∞ ∞
86 6631 ∞ ∞
95 7332 ∞ ∞
104 8133 ∞ ∞
118 9034 ∞ ∞
129 99Note, that all items under an ∞ in a column also are ∞ . In the sequel, in each column,just the top ∞ (if any) is printed. n . In the table, a single value indicates that this is the exact e-value. Two values separatedby a dash (–) are the best known lower and upper bounds of the respective e-value. Again, l = k + 1. n \ l ∞ ∞ ∞ ∞ By hand calculations or by means of e. g. the matlab programme FRANK ([6]) , itis fairly easy to check for consequences for upper bounds on Ramsey numbers for anyimprovement of lower bounds of e-numbers. As compared to the combined values from[4] and older versions of [1], the sharper bounds presented here yield just two improvedupper Ramsey number bounds.It turned out that the improvement of the lower bound for e(3 ,
12; 43) from 128 to 129was crucial for deducing that R(3 , , as reported in the latest dynamic survey on small Ramsey numbers ([8]).The improvement of lower e(3 ,
11; 39) bound from 117 ([4]) to 119 suffices to provethat R(3 , . This bound is not (yet) included in the dynamic survey.
Most of the ‘sporadic’ lower bounds are found in [4]; and/or are direct consequences oflower bounds for smaller independence numbers. The exceptions are the lower bounds fore(3 ,
11; 35), e(3 ,
12; 38), e(3 ,
12; 39), e(3 ,
13; 41), e(3 ,
13; 42), e(3 ,
12; 43), e(3 ,
11; 39), ande(3 ,
11; 41).The first six of these bounds, as well as the ‘general’ bounds, depend partly on the-oretical classification of some ‘lower’ graphs, i. e., graphs with lower independence andvertex numbers; likewise, the two last ones depend on computational classification of somelower graphs. In all cases, there is some use of properties deduced for some lower graphs;and the general proof technique is to assume the existence of a graph G with ‘offendingly’low e ( G ), and then to deduce more and more precise conditions for G , until finally acontradiction is achieved. I’ll provide a few examples.First, assume that G is a (3,11;35)-graph with e ( G )
83; whence actually equalitymust hold. We then successively may prove: The version of FRANK that I employ includes a test for raising the lower e-number bound in a fewcases, where the only formally possible degree distributions all would have to contain either a triangleof low-degree vertices, or a low-degree vertex with too few low-degree neighbours (and thus a too highsecond degree). In practice, this only may happen, when the unraised e-number bound would be closeto, but slightly less than, the e-value for some regular graph. This tweak yielded e. g. e(3 ,
13; 51) > a ) δ ( G ) > b ) δ ( G ) > c ) any vertex of degree 4 has at most one neighbour of degree > d ) G v has no W , component for any vertex of degree 5; and( e ) if deg( v ) = 5, then deg ( v ) a ) is immediate from the e(3 , n ) values.( b ) follows from ( a ), and from the fact that any (3 ,
10; 31)-graph H with e ( H ) δ ( H ) >
2, strictly if e ( H ) = 73; and that there are at most two vertices of degree 2in H , which (if indeed there are two of them) moreover must be adjacent.( c ) is immediate from ( b ), and the fact that deg ( v )
17 for any vertex of degree 4.( e ) is an immediate consequence of ( d ), and of the fact that any (3 ,
10; 29 , W , component. On the other hand, ( e ) directly yields a contradiction,since it means that we could calculate as if e(3 ,
10; 29) were at least 59.This just leaves the deduction of ( d ) from ( b ) and ( c ), which is somewhat less imme-diate. Assume for a contradiction that deg( v ) = 5, and that G v has a W , component.Let N ( v ) = { w , . . . , w } , and let U be the set of vertices in W , , which are not adjacentto any w i ; in other words, U = { u ∈ V ( W , ) : deg G ( u ) = 4 } .Now, | U |
8, since U cannot contain an independent 4-set; if it did, any edge between U and N ( v ) would be redundant (in the sense that removing it from G would leave a graphwhich also did not contain an independent 11-set), but G can contain neither a redundantedge, nor a W , component. Thus, and by inspection of W , , if U were non-empty,then there were a u j ∈ U with at most two neighbours in U , and therefore at least twoneighbours of degrees >
5, contradicting ( c ).Thus, instead, U = ∅ ; i. e., each vertex in W , is adjacent to at least one w i . Thismakes it possible to apply a “decharging” argument. ‘Charge’ each u j with a unit charge,1; and then ‘discharge’ each u j by distributing its charge in equal proportions to its w i neighbours. The total charge after discharging must stay 13. However, no w i can receivea charge larger than 2 .
5; which means that N ( v ) in total cannot carry a higher chargethan 12 .
5. This is a contradiction; which indeed proves ( d ).For a second example, assume that G is a (3 ,
11; 41)-graph with e ( G ) = 138. Thereare few theoretic ways for such a graph to be ‘realised numerically’; in other words, if welet the degree distribution (degree sequence) of the graph be ( n , n , . . . , n ), then thereare just a handful possible such sequences, for which the resulting Graver-Yackel defect γ ( G ) would be non-negative (cf. [5] and [4]). In fact, also employing that a single vertex v of degree 8 would have deg ( v ) · , , (12 , , , (1 , , , (2 , , , and (3 , , γ ( G ) = 3, 1, 2, 1, and 0, respectively.Put F := { v ∈ V : deg( v ) = 7 and deg ( v ) = 48 } . In other words, F is the set ofnon-defect vertices of degree 7. Counting directly yields that | F | >
27, in each one of thecases. 6or any f ∈ F , G f is a (3 ,
10; 33 , House ofGraphs ([4]). Running the NAUTY ([7]) command countg --Jd on this list reveals thatany such graph H contains an induced K , , and has δ ( H ) >
4. Moreover, a theoreticalanalysis shows that for any vertex v with 5 deg( v )
7, either δ ( G v ) >
3, or δ ( G v ) = 2and γ ( v ) = 3, or γ ( v ) > f ; if there is a vertex x of degree 8, actually choose f ∈ F ∩ N ( x );choose a K , ⊂ V f ⊂ V , with V ( K , ) = { a , a ; b , . . . , b } and deg( a ) deg( a ) δ ( G a i ) deg( a ) − i − , for i = 1 , a i .If deg( a ) = 5, then γ ( a ) > > > γ ( G ), a contradiction. Likewise, if deg( a ) = 6,then γ ( a ) = 3, whence then γ ( a ) = 0; whence anyhow6 deg( a ) a ) . If deg( a ) = 7, then both a and a are defective, and the further defects in G sum up toat most 1, whence in particular then ∆( G ) = 7. Moreover, if deg( a ) = 7, then not both a and a may have defects >
2, whence instead then at least one of them has secondvalency 47, and thus at least five neighbours of degree 7, of which at least four belong to F . Thus, in this case, we may assume that f ′ := b ∈ F ; while if deg( a ) = 6, then let f ′ be arbitrarily chosen in F ∩ lk a . In either case, there is some K , in V f ′ , and thiswould also carry a defect at least 2, which would yield a total defect at least 4 in G , acontradiction. For n k = 4 l − n (but excepting ( n, l ) ∈ { (17 , , (22 , , (27 , } ), there are con-structions, whose connected components either are described by their extension patterns,or are one or the other of two exceptional graphs : The cyclic graph W , (the unique(3,5;13,26)-graph), and the twisted tesseract (a (3,6;16,32)-graph). (The twisted tesseractalso is denoted (2 W , ) i in [1]; i. e., it consists of two disjoint copies of W , , with the i ’th vertex in the first copy connected to the 5 i ’th one in the second copy by an edge;where indices are taken modulo 8.)The extension pattern of a graph G of the kind we consider here includes a trianglefree graph T , such that e ( T ) n ( T ) ,α ( G ) = n ( T ) ,n ( G ) = 2 n ( T ) + e ( T ) , and e ( G ) = n ( T ) + 2 e ( T ) + 12 X x ∈ V ( T ) deg( x ) . k and n with 3 . k n k , we have suchgraphs realising equality in (1). However, there are some irregularities, for two reasons.First, each W , component contributes 4 to the independence number of the graph;and there may not be an integer number of such components that realises equality in (1).Second, in general, for a connected patterned graph G with 3 . α ( G ) n ( G ) α ( G ),equality only can be achieved by having only vertices of degrees 3 and 4 in the patterngraph T (since other degree distributions yield higher P V ( T ) deg( x ) ); which for (3 ,
10; 36)-graphs would force the pattern graph to be 4-regular, on 9 vertices. By inspection, thereis no such triangle-free graph; the closest possible degree distribution is (2,5,2) vertices ofdegrees (3,4,5), respectively.The upper bound 161 for e(3 ,
10; 39) is reported by Goedgebeur and Radziszowski in[4], where it is noted that both they and Exoo have found huge amounts of (3 ,
10; 39 , G , but no (3 ,
10; 39)-graph with a lower number of edges.For the five upper bounds within parentheses, let L be the regular (3 )-type lace withconstant offsets (1,3), a (3 ,
9; 32 , v , . . . , v ) of apices consistsof non-adjacent vertices of degree 6, where moreover dist( v i , v j ) >
3, if i and j have thesame parity. The upper e(3 ,
10; 37) (e(3 ,
10; 38)) bounds are achieved by a 4-extension(5-extension) of H , employing 3 (all 4) of the odd-indexed v i , respectively; and the uppere(3 ,
11; 41—43) bounds by making a further extension of one of these, employing the v i with even indices. References [1] J. Backelin, Contributions to a Ramsey calculus, unpublished manuscript .[2] J. Backelin, Edge number critical triangle-free graphs with low independence numbers, arXiv:1309.7874 ( unpublished New computational upper bounds for Ramseynumbers R (3 , k ), Electronic Journal of Combinatorics
J. Comb. Theory , Series
A;4 (1968), 125–175.[6] A. Lesser, Theoretical and computational aspects of Ramsey theory,
Examensarbeteni Matematik , Matematiska Institutionen, Stockholms Universitet (2001).[7] B. D. McKay, A. Piperno, Practical graph isomorphism, II J. Symbolic Computa-tion (2014), 94–112; doi:10.1016/j.jsc.2013.09.003 .[8] S. P. Radziszowski, Small Ramsey numbers; Dynamic Survey, Revision Elec-tronic Journal of Combinatorics (2014) 89] S. P. Radziszowski and D. L. Kreher, On (3,k) Ramsey graphs: Theoretical andcomputational results,
J. Comb Math. and Comb. Computing (1988), 37–52.[10] S. P. Radziszowski and D. L. Kreher, Minimum triangle-free graphs, Ars Comb .31