Abstract
AbstractIn this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $$K_{m,n}$$
K
m
,
n
, extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that $$\mathop {\textrm{cr}}\limits (K_{10,n}) \ge 4.87057 n^2 - 10n$$
cr
(
K
10
,
n
)
≥
4.87057
n
2
-
10
n
, $$\mathop {\textrm{cr}}\limits (K_{11,n}) \ge 5.99939 n^2-12.5n$$
cr
(
K
11
,
n
)
≥
5.99939
n
2
-
12.5
n
, $$ \mathop {\textrm{cr}}\limits (K_{12,n}) \ge 7.25579 n^2 - 15n$$
cr
(
K
12
,
n
)
≥
7.25579
n
2
-
15
n
, $$\mathop {\textrm{cr}}\limits (K_{13,n}) \ge 8.65675 n^2-18n$$
cr
(
K
13
,
n
)
≥
8.65675
n
2
-
18
n
for all n. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.
Publisher
Springer Science and Business Media LLC
Subject
General Mathematics,Software
Reference28 articles.
1. Balogh, J., Lidický, B., Salazar, G.: Closing in on Hill’s conjecture. SIAM J. Discrete Math. 33, 1261–1276 (2019)
2. Balogh, J., Lidický, B., Norin, S., Pfender, F., Salazar, G., Spiro, S.: Crossing numbers of complete bipartite graphs. Procedia Computer Science 223, 78–87 (2023)
3. Brosch, D.: Symmetry reduction in convex optimization with applications in combinatorics. Ph.D. thesis, Tilburg University (2022)
4. Cameron, P.J.: Permutation Groups. Cambridge University Press, Cambridge (1999)
5. de Klerk, E., Maharry, J., Pasechnik, D.V., Richter, R.B., Salazar, G.: Improved bounds for the crossing numbers of $$K_{m, n}$$ and $$K_n$$. SIAM J. Discrete Math. 20, 189–202 (2006)