Generalized Ramsey numbers through adiabatic quantum optimization

Abstract

Ramsey theory is an active research area in combinatorics whose central theme is the emergence of order in large disordered structures, with Ramsey numbers marking the threshold at which this order first appears. For generalized Ramsey numbers r(G, H), the emergent order is characterized by graphs G and H. In this paper we: (i) present a quantum algorithm for computing generalized Ramsey numbers by reformulating the computation as a combinatorial optimization problem which is solved using adiabatic quantum optimization; and (ii) determine the Ramsey numbers $r({\mathscr {T}}_{m},{\mathscr {T}}_{n})$ for trees of order $m,n = 6,7,8$, most of which were previously unknown.

Appendices

Appendix 1: Tree Ramsey numbers: literature survey

In this Appendix we quote five theorems from the literature pertaining to generalized Ramsey numbers for certain families of trees. These theorems: (i) allow us to identify which of our tree Ramsey numbers are new and (ii) provide checks on the remainder of our results. Section 2 provides a brief review of the graph theory concepts needed in this paper.

Theorem 1

(Gerecsér and Gyárfás [20]) For paths $P_{m}$ and $P_{n}$ with $2\le m\le n$,

$$\begin{aligned} r(P_{m},P_{n}) = n + \bigg \lfloor \frac{m}{2}\bigg \rfloor - 1. \end{aligned}$$

Theorem 2

(Harary [21] ) For stars $K_{1,m-1}$ and $K_{1,n-1}$ with $m,n \ge 2$ and diameter 2,

$$\begin{aligned} r(K_{1,m-1}, K_{1,n-1}) = \left\{ \begin{array}{ll} m+n-3, &{} \mathrm {odd}\; m,n\\ m+n -2, &{} \mathrm {otherwise.} \end{array} \right. \end{aligned}$$

Theorem 3

(Cockayne [22]) If $T_{m}$ is a tree of order m containing a vertex of degree one adjacent to a vertex of degree two, then $r(T_{m},K_{1,n-1}) = m+n-3$ ($n\ge 2$), provided one of the following holds:

• 1. $n - 1 \equiv 0,2 \;\bmod (m-1)$ ;

• 2. $n - 1 \not \equiv 1 \;\bmod (m-1)$ and $n-1 \ge (m-3)^{2}$;

• 3. $n-1 \not \equiv 1 \;\bmod (m-1)$ and $n-1 \equiv 1 \;\bmod (m-2)$;

• 4. $n-1 \equiv m-2 \;\bmod (m-1)$ and $n-1 > m-2$.

The following definition proves convenient.

Definition 2

Suppose $a,b\ge 1$ and $k\ge 2$. Then $S^{(k)}_{a,b}$ is the graph obtained from the disjoint stars $K_{1,a}$ and $K_{1,b}$ by joining their centers with a path of order k (viz. length $k-1$).

Thus the order of $S^{(k)}_{a,b}$ is $a+b+k$ and the path $P_{n} = S^{(n-2)}_{1,1}$ for $n\ge 4$.

Theorem 4

(Burr and Erdös [23]) Consider the graph $S^{(4)}_{a,b}$ with $a\ge b\ge 1$ and diameter 5. Then

$$\begin{aligned} r(S^{(4)}_{a,b},S^{(4)}_{a,b}) = \max \{ 2a+3,a+2b+5\} . \end{aligned}$$

Theorem 5

(Grossman et al. [24]) Consider the graph $S^{(2)}_{a,b}$ with $a\ge b\ge 1$ and diameter 3. Then

$$\begin{aligned} r(S^{(2)}_{a,b},S^{(2)}_{a,b}) \ge \left\{ \begin{array}{ll} \max \{ 2a+1,a+2b+2\}, &{} \quad a\;\mathrm {odd, and}\; b = 1,2\\ \max \{ 2a+2,a+2b+2\}, &{} \quad \mathrm {otherwise.} \end{array} \right. \end{aligned}$$

Furthermore,

$$\begin{aligned} r(S^{(2)}_{a,b},S^{(2)}_{a,b}) = \left\{ \begin{array}{ll} \max \{ 2a+1,a+2b+2\}, &{} \quad a\;\mathrm {odd, and}\; b = 1,2\\ \max \{ 2a+2,a+2b+2\}, &{} \quad a\;\mathrm {even, or}\; b\ge 3\; \mathrm {provided}\; a\le \sqrt{2}b\;\mathrm {or}\; a\ge 3b. \end{array} \right. \end{aligned}$$

Appendix 2: Remaining tree Ramsey numbers

In this Appendix we present our remaining tree Ramsey number results. “Tree Ramsey numbers $r({\mathscr {T}}_{7},{\mathscr {T}}_{n})$ for $n=6,7$” Appendix section contains our results for $r({\mathscr {T}}_{7},{\mathscr {T}}_{n})$ with $n=6,7$; and “$r({\mathscr {T}}_{8},{\mathscr {T}}_{n})$ for $n = 6,7,8$ Appendix” section contains $r({\mathscr {T}}_{8},{\mathscr {T}}_{n})$ for $n = 6,7,8$. “Tree Ramsey numbers $r({\mathscr {T}}_{7},{\mathscr {T}}_{n})$ for $n=6,7$” Appendix section also present, for each input pair of GI classes, the number of non-isomorphic critical graphs, and at the Ramsey threshold $N = r({\mathscr {T}}^{i}_{m},{\mathscr {T}}_{n}^{j})$, the number of non-isomorphic optimal graphs and their associated minimum objective function values.

1.1 Tree Ramsey numbers $r({\mathscr {T}}_{7},{\mathscr {T}}_{n})$ for $n=6,7$

Trees of order 7 partition into eleven GI classes [18] which we denote by $\{ {\mathscr {T}}^{j}_{7} : j = 1,\ldots 11\}$ and show as unlabeled graphs in Fig. 2. Seven of these GI classes correspond to known (unlabeled) graphs (Sect. 2; “Appendix 1”):

$$\begin{aligned} \begin{array}{lll} {\mathscr {T}}_{7}^{1} = P_{7} = S^{(5)}_{1,1}; &{} \quad {\mathscr {T}}_{7}^{6} = S^{(3)}_{3,1}; &{} \quad {\mathscr {T}}_{7}^{10} = K_{1,6}.\\ {\mathscr {T}}_{7}^{2} = S^{(4)}_{2,1} ; &{} \quad {\mathscr {T}}_{7}^{8} = S^{(2)}_{3,2};&{} \\ {\mathscr {T}}_{7}^{5} = S^{(3)}_{2,2}; &{} \quad {\mathscr {T}}_{7}^{9} = S^{(2)}_{4,1};&{} \end{array} \end{aligned}$$

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Fig. 2

Graph isomorphism classes ${\mathscr {T}}^{j}_{7}$ ($j=1,\ldots 11$) for trees of order 7. a ${\mathscr {T}}^{1}_{7}$. b ${\mathscr {T}}^{2}_{7}$. c ${\mathscr {T}}^{3}_{7}$. d ${\mathscr {T}}^{4}_{7}$. e ${\mathscr {T}}^{5}_{7}$. f ${\mathscr {T}}^{6}_{7}$. g ${\mathscr {T}}^{7}_{7}$. h ${\mathscr {T}}^{8}_{7}$. i ${\mathscr {T}}^{9}_{7}$. j ${\mathscr {T}}^{10}_{7}$. k ${\mathscr {T}}^{11}_{7}$

1.1.1 $r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j})$

Using our numerical procedure (Sect. 4.1) we determined the tree Ramsey numbers $r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j})$ for $1\le i\le 11$ and $1\le j\le 6$ which are displayed in Table 6. A superscript “x” on a table entry indicates that Theorem A.x of “Appendix 1” applies, and so these tree Ramsey numbers were known prior to this work. The reader can verify that our numerical results are in agreement with the theorems of “Appendix 1”. The remaining 64 tree Ramsey numbers (to the best of our knowledge) are new.

Table 6 Numerical results for tree Ramsey numbers $r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j})$ with $1\le i \le 11$ and $1\le j \le 6$

As explained in Sect. 4.1, for trees of order 6 and 7, and for graphs of order $8\le N\le 11$, our numerical procedure can exhaustively search over all non-isomorphic graphs, and so can find the number of non-isomorphic optimal graphs and their associated objective function value. For graphs with order $N = r({\mathscr {T}}_{6}^{i},{\mathscr {T}}_{6}^{j}) -1$, the optimal graphs are (${\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j}$)-critical graphs (Sect. 2). Table 7 lists the number of non-isomorphic critical graphs for each pairing of GI classes $({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j})$. These graphs are all found to have vanishing objective function value, which is expected, since critical graphs have order $N < r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j})$.

Table 7 Numerical results for the number of non-isomorphic (${\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j}$)-critical graphs $N_\mathrm{c}({\mathscr {T}}_{7}^{i}, {\mathscr {T}}_{6}^{j})$ with $1\le i\le 11$ and $1\le j \le 6$

At the Ramsey threshold, $N = r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j})$, optimal graphs first acquire a non-vanishing objective function value (see Sect. 3.1). In Tables 8 and 9 we list, respectively, the number of non-isomorphic optimal graphs for each pairing of GI classes $({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j})$ and the corresponding minimum objective function value.

Table 8 Numerical results for the number of non-isomorphic optimal graphs $N_\mathrm{opt}({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j})$ with $1\le i \le 11$ and $1\le j \le 6$

Table 9 Numerical results for the minimum objective function value ${\mathscr {O}}_{N}(e_{*};{\mathscr {T}}_{7}^{i},{\mathscr {T}}_{6}^{j})$ at the Ramsey threshold with $1\le i \le 11$ and $1\le j \le 6$

Table 10 Numerical results for tree Ramsey numbers $r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$ with $1\le i,j \le 11$

1.1.2 $r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$

Using the numerical procedure described in Sect. 4.1, we determined the tree Ramsey numbers $r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$ for $1\le i,j\le 11$ which are displayed in Table 10. Only the upper triangular table entries are shown as the lower triangular entries follow from $r({\mathscr {T}}_{7}^{j},{\mathscr {T}}_{7}^{i}) =r({\mathscr {T}}_{7}^{i}, {\mathscr {T}}_{7}^{j})$. A superscript “x” on a table entry indicates that Theorem A.x in “Appendix 1” applies, and so these tree Ramsey numbers were known prior to this work. The reader can verify that our numerical results are in agreement with the theorems of “Appendix 1”. The remaining 102 tree Ramsey numbers (to the best of our knowledge) are new.

As explained in Sect. 4.1, for $m,n=7$, and for graphs of order $9\le N\le 11$, our numerical procedure can exhaustively search over all non-isomorphic graphs, and so can find the number of non-isomorphic optimal graphs and their associated objective function value. For graphs with order $N = r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j}) -1$, the optimal graphs are (${\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j}$)-critical graphs (Sect. 2). Table 11 lists the number of non-isomorphic critical graphs for each pairing of GI classes $({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$. These graphs are all found to have vanishing objective function value, which is expected, since critical graphs have order $N < r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$.

Table 11 Numerical results for the number of non-isomorphic critical graphs $N_\mathrm{c}({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$ with $1\le i,j \le 11$

At the Ramsey threshold, $N = r({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$, optimal graphs first acquire a non-vanishing objective function value (see Sect. 3.1). In Tables 12 and 13 we list, respectively, the number of non-isomorphic optimal graphs for each pairing of GI classes $({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$ and the corresponding minimum objective function value.

Table 12 Numerical results for the number of non-isomorphic optimal graphs $N_\mathrm{opt}({\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$ with $1\le i,j \le 11$

Table 13 Numerical results for the minimum objective function value ${\mathscr {O}}_{N}(e_{*};{\mathscr {T}}_{7}^{i},{\mathscr {T}}_{7}^{j})$ at the Ramsey threshold with $1\le i,j \le 11$

1.2 $r({\mathscr {T}}_{8},{\mathscr {T}}_{n})$ for $n = 6,7,8$

Trees of order 8 partition into twenty-three GI classes [18] which we denote by $\{ {\mathscr {T}}^{j}_{8} : j = 1,\ldots 23\}$, and show as unlabeled graphs in Fig. 3. Ten of these GI classes correspond to known (unlabeled) graphs (Sect. 2; “Appendix 1”):

$$\begin{aligned} {\mathscr {T}}_{8}^{1}= & {} K_{1,7}; \qquad \qquad {\mathscr {T}}_{8}^{4} = S^{(2)}_{4,2}; \qquad \qquad {\mathscr {T}}_{8}^{9} = S^{(4)}_{3,1}; \qquad \qquad {\mathscr {T}}_{8}^{23} = P_{8} = S^{(6)}_{1,1}. \nonumber \\ {\mathscr {T}}_{8}^{2}= & {} S^{(2)}_{5,1}; \qquad \qquad {\mathscr {T}}_{8}^{5} = S^{(3)}_{3,2}; \qquad \qquad {\mathscr {T}}_{8}^{12} = S^{(4)}_{2,2}; \nonumber \\ {\mathscr {T}}_{8}^{3}= & {} S^{(3)}_{4,1}; \qquad \qquad {\mathscr {T}}_{8}^{6} = S^{(2)}_{3,3}; \qquad \qquad {\mathscr {T}}_{8}^{14} = S^{(5)}_{2,1}; \end{aligned}$$

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Fig. 3

Graph isomorphism classes ${\mathscr {T}}^{j}_{8}$ ($j=1,\ldots 23$) for trees of order 8. a ${\mathscr {T}}^{1}_{8}$. b ${\mathscr {T}}^{2}_{8}$. c ${\mathscr {T}}^{3}_{8}$. d ${\mathscr {T}}^{4}_{8}$. e ${\mathscr {T}}^{5}_{8}$. f ${\mathscr {T}}^{6}_{8}$. g ${\mathscr {T}}^{7}_{8}$. h ${\mathscr {T}}^{8}_{8}$. i ${\mathscr {T}}^{9}_{8}$. j ${\mathscr {T}}^{10}_{8}$. k ${\mathscr {T}}^{11}_{8}$. l ${\mathscr {T}}^{12}_{8}$. m ${\mathscr {T}}^{13}_{8}$. n ${\mathscr {T}}^{14}_{8}$. o ${\mathscr {T}}^{15}_{8}$. p ${\mathscr {T}}^{16}_{8}$. q ${\mathscr {T}}^{17}_{8}$. r ${\mathscr {T}}^{18}_{8}$. s ${\mathscr {T}}^{19}_{8}$. t ${\mathscr {T}}^{20}_{8}$. u ${\mathscr {T}}^{21}_{8}$. v ${\mathscr {T}}^{22}_{8}$. w ${\mathscr {T}}^{23}_{8}$

1.2.1 $r({\mathscr {T}}_{8}^{i},{\mathscr {T}}_{6}^{j})$

The numerical procedure described in Sect. 4.1 was used to determine the tree Ramsey numbers $r({\mathscr {T}}_{8}^{i},{\mathscr {T}}_{6}^{j})$ for $1\le i\le 23$ and $1\le j\le 6$, and the results are displayed in Table 14. A superscript “x” on a table entry indicates that Theorem A.x of “Appendix 1” applies, and so these tree Ramsey numbers were known prior to this work. The reader can verify that our numerical results are in agreement with the theorems of “Appendix 1”. Recall from Sect. 4.1 that for graphs with order $N\ge 12$, exhaustive search over non-isomorphic graphs was not feasible. For such graphs, we used the heuristic algorithm Tabu search to look for minima of the tree Ramsey number objective function. As noted there, Tabu search only yields a lower bound for a tree Ramsey number. However, because of Theorem A.3, we know that for $r({\mathscr {T}}_{8}^{1},{\mathscr {T}}_{6}^{5})$, the value 12 is in fact the actual value of this Ramsey number. Thus no lower bounds appear in Table 14—all values are actual Ramsey number values. Of the 138 tree Ramsey numbers appearing in Table 14, (to the best of our knowledge) 132 are new.

Table 14 Numerical results for tree Ramsey numbers $r({\mathscr {T}}_{8}^{i},{\mathscr {T}}_{6}^{j})$ with $1\le i \le 23$ and $1\le j \le 6$

1.2.2 $r({\mathscr {T}}_{8}^{i},{\mathscr {T}}_{7}^{j})$

The numerical results for the tree Ramsey numbers $r({\mathscr {T}}_{8}^{i}, {\mathscr {T}}_{7}^{j})$ for $1\le i\le 23$ and $1\le j\le 11$ are shown in Table 15. A superscript “x” on a table entry indicates that Theorem A.x of “Appendix 1” applies, and so these tree Ramsey numbers were known prior to this work. The reader can verify that our numerical results are in agreement with the theorems of “Appendix 1”. Recall from Sect. 4.1 that for graphs with order $N\ge 12$, exhaustive search over non-isomorphic graphs was not feasible. For such graphs we used the heuristic algorithm Tabu search to look for minima of the tree Ramsey number objective function. As noted there, Tabu search only yields a lower bound for a tree Ramsey number. Numbers in Table 15 marked with an asterisk correspond to lower bounds on the associated tree Ramsey number. Of the 253 numbers appearing in this table (to the best of our knowledge), 241 are new tree Ramsey numbers, 10 are lower bounds, and 2 were previously known.

Table 15 Numerical results for tree Ramsey numbers $r({\mathscr {T}}_{8}^{i},{\mathscr {T}}_{7}^{j})$ with $1\le i \le 23$ and $1\le j \le 11$

Table 16 Numerical results for tree Ramsey numbers $r({\mathscr {T}}_{8}^{i},{\mathscr {T}}_{8}^{j})$ with $1\le i,j \le 23$

Before moving on we note that it has been conjectured [19] that $r({\mathscr {T}}_{m},{\mathscr {T}}_{n})\le n + m - 2$ for all trees ${\mathscr {T}}_{m}$ and ${\mathscr {T}}_{n}$. For the trees in Table 15, the conjecture gives the upper bound $r({\mathscr {T}}_{8},{\mathscr {T}}_{7})\le 13$. The conjecture is seen to be consistent with all entries in Table 15. Furthermore, if the conjecture is true, then all lower bounds in row 1 become exact values since we would have $13\le r({\mathscr {T}}_{8}^{1}, {\mathscr {T}}_{7}^{j}) \le 13$.

1.2.3 $r({\mathscr {T}}_{8}^{i},{\mathscr {T}}_{8}^{j})$

The numerical results for the tree Ramsey numbers $r({\mathscr {T}}_{8}^{i}, {\mathscr {T}}_{8}^{j})$ for $1\le i,j\le 23$ are shown in Table 16. Only the upper triangular table entries are shown as the lower triangular entries follow from $r({\mathscr {T}}_{8}^{j},{\mathscr {T}}_{8}^{i}) =r({\mathscr {T}}_{8}^{i}, {\mathscr {T}}_{8}^{j})$. A superscript “x” on a table entry indicates that Theorem A.x in “Appendix 1” applies, and so these tree Ramsey numbers were known prior to this work. The reader can verify that our numerical results are in agreement with the theorems of “Appendix 1”. Recall from Sect. 4.1 that for graphs with order $N\ge 12$, exhaustive search over non-isomorphic graphs was not feasible. For such graphs we used the heuristic algorithm Tabu search to look for minima of the tree Ramsey number objective function. As noted there, Tabu search only yields a lower bound for a tree Ramsey number. Numbers in Table 16 marked with an asterisk correspond to lower bounds on the associated tree Ramsey number. Of the 529 numbers appearing in this table, (to the best of our knowledge) 479 are new tree Ramsey numbers, 10 are lower bounds, and 40 were previously known. The conjectured upper bound [19], $r({\mathscr {T}}_{m},{\mathscr {T}}_{n})\le n + m - 2$, evaluates to $r({\mathscr {T}}_{8}, {\mathscr {T}}_{8})\le 14$ and is seen to be consistent with all entries in Table 16.