Paul Erdős suggested the following problem: Determine or estimate the number of maximal triangle‐free graphs on n vertices. Here we show that the number of maximal triangle‐free graphs is at most 2n2/8+o(n2), which matches the previously known lower bound.
Our proof uses among others the Ruzsa–Szemerédi triangle‐removal lemma, and recent results on characterizing of the structure of independent sets in hypergraphs.
Recently, settling a question of Erdős, Balogh, and Petříčková showed that there are at most 2 n 2 /8+o(n 2 ) n-vertex maximal triangle-free graphs, matching the previously known lower bound. Here, we characterize the typical structure of maximal triangle-free graphs. We show that almost every maximal triangle-free graph G admits a vertex partition X ∪ Y such that G[X ] is a perfect matching and Y is an independent set.Our proof uses the Ruzsa-Szemerédi removal lemma, the Erdős-Simonovits stability theorem, and recent results of Balogh, Morris, and Samotij and Saxton and Thomason on characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint P 3 s, which is of independent interest.
Abstract. Recently, Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some k such that all kth power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant c such that for any k there is a family of graphs G with χ(G k ) unbounded and χ (G k ) ≥ cχ(G k ) log χ(G k ). We also provide an upper bound, χ (G k ) < χ(G k ) 3 for k > 1.
Given a poset P we say a family F ⊆ P is centered if it is obtained by 'taking sets as close to the middle layer as possible'. A poset P is said to have the centeredness property if for any M , among all families of size M in P , centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice {0, 1} n has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset {0, 1, . . . , k} n also has the centeredness property, provided n is sufficiently large compared to k. We show that this conjecture is false for all k ≥ 2 and investigate the range of M for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of F n q has the centeredness property. Several open questions are also given.
An online Ramsey game $(G,\mathcal{H})$ is a game between Builder and Painter, alternating in turns. During each turn, Builder draws an edge, and Painter colors it blue or red. Builder's goal is to force Painter to create a monochromatic copy of $G$, while Painter's goal is to prevent this. The only limitation for Builder is that after each of his moves, the resulting graph has to belong to the class of graphs $\mathcal{H}$. It was conjectured by Grytczuk, Hałuszczak, and Kierstead (2004) that if $\mathcal{H}$ is the class of planar graphs, then Builder can force a monochromatic copy of a planar graph $G$ if and only if $G$ is outerplanar. Here we show that the "only if" part does not hold while the "if" part does.
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