18、Token Jumping in Kℓ,ℓ-Free Graphs: A Polynomial Kernel Approach

Token Jumping in Kℓ,ℓ-Free Graphs: A Polynomial Kernel Approach

1. Introduction to Kℓ,ℓ-Free Graphs

Kℓ,ℓ-free graphs are graphs that do not contain a complete bipartite sub - graph (K_{\ell,\ell}). Kővári et al. proved that such graphs have a sub - quadratic number of edges.

• Theorem 2 : For a (K_{\ell,\ell}) - free graph (G) with (n) vertices, the number of edges is at most (ex(n, K_{\ell,\ell})), where (ex(n, K_{\ell,\ell})\leq\left\lfloor\frac{\ell - 1}{2}\right\rfloor^{1/\ell}\cdot n^{2 - 1/\ell}+\frac{1}{2}(\ell - 1)n).
• Corollary 1 : Every (K_{\ell,\ell}) - free graph with (k^{\ell}(4k)^{\ell}) vertices contains an independent set of size (k).
• Theorem 3 (Bipartite Version