The path–biclique conjecture for bounded alpha-degeneracy

Let $P_t$ denote the $t$-vertex path and $K_{,}$ the complete bipartite graph with

vertices in each part. For a graph $G$, its **\alpha-degeneracy** is the smallest integer $k$ such that every non-null induced subgraph $H$ of $G$ contains a vertex $v$ with

. A graph is

$$-free if it has no induced subgraph isomorphic to$$

or $K_{,}$. Path–biclique alpha-degeneracy conjecture. For every two positive integers $t$ and

$$, the class of$$

-free graphs has bounded alpha-degeneracy.

Alpha-degeneracy is a lower bound on tree-independence number, so this conjecture is a necessary-condition analogue of the corresponding tree-independence conjecture. The source does not report a resolution of the general statement; the case $t=5$ is not asserted here as a separate result for alpha-degeneracy.