A clique-coloring of a given graph G is a coloring of the vertices of G such that no maximal clique of size at least two is monocolored. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. It has been proved that every planar graph is 3-clique colorable and every claw-free planar graph, different from an odd cycle, is 2-clique colorable. In this ...

Mader conjectured that every C4-free graph has a subdivision of a clique of order linear in its average degree. We show that every C6-free graph has such a subdivision of a large clique. We also prove the dense case of Mader’s conjecture in a stronger sense, i.e. for every c, there is a c′ such that every C4-free graph with average degree cn 1/2 has a subdivision of a clique K` with ` = bc′n1/2...

A claw-free graph G(V,E) is said to be basic if there exists a matching M ⊆ E whose edges are strongly bisimplicial and such that each connected component C of G − M is either a clique or a {claw, net}-free graph or satisfies α(G[C \ V (M)]) ≤ 3. The Maximum Weight Stable Set (MWSS) Problem in a basic claw-free graph can be easily solved by first solving at most four MWSS problems in each conne...

Abstract We propose a novel graph-based approach for semantic parsing that resolves two problems observed in the literature: (1) seq2seq models fail on compositional generalization tasks; (2) previous work using phrase structure parsers cannot cover all parses treebanks. prove both MAP inference and latent tag anchoring (required weakly-supervised learning) are NP-hard problems. optimization al...