In 1966, Erd\H{o}s, Goodman, and P\'{o}sa showed that every $n$-vertex graph $G$ admits a cover of its edges using at most $\lfloor \frac{n^2}{4}\rfloor$ cliques, with tightness witnessed by the balanced complete bipartite graph. Erd\H{o}s suggested the following strengthening: every $n$-vertex graph $G$ admits an edge decomposition into cliques with total cost at most $\lfloor \frac{n^2}{4}\rfloor$, where each $i$-clique has cost $i-1$. There is another generalization conjectured by Dau, Milenkovic and Puleo: for every $t\ge4$, every $n$-vertex graph $G$ admits a cover of its $t$-cliques using at most $\prod_{j=0}^{t-1}\left\lfloor\frac{n+j}{t}\right\rfloor$ cliques. Balogh, He, Krueger, Nguyen and Wigal proved asymptotic and fractional versions of these conjectures. We solve both conjectures for large $n$ and identify the equality cases. Our proof of Erd\H{o}s's conjecture combines linear programming duality and the spanning-forest polytope with a weighted stability form of Mantel's theorem, followed by explicit decomposition algorithms around an almost balanced bipartition. For the $t$-clique-cover conjecture, we combine graph removal lemma and generalized Tur\'an stability with an exact covering construction for graphs close to $T_{n,t}$.
For a graph $G$, let $\text{cg}(G)$ be the maximum $k\in \mathbb N$ for which $G$ contains a $k$-vertex cograph as an induced subgraph. The Erd\H{o}s-Hajnal conjecture asserts that for every proper hereditary graph class $\mathcal{G}$, there exists $\gamma\in(0,1]$ such that every $n$-vertex graph $G\in\mathcal{G}$ sat...
For a fixed finite collection of graphs $\mathcal{F}$, the $\mathcal{F}$-Minor-Free Deletion problem asks, given a graph $G$ and an integer $k$, whether there is a set $S\subseteq V(G)$ with $|S|\leq k$ such that $G-S$ contains no graph of $\mathcal{F}$ as a minor. This generalizes Vertex Cover, Feedback Vertex Set, an...
Jongseo Lee, Roohani Sharma, Sebastian Wiederrecht· 0 citations
For a graph $H$, the suspension $\widehat H$ of $H$ is a graph obtained from $H$ by joining a new vertex to every vertex of $H$. The generalized Tur\'an number $\textrm{ex}(n,K_3,\widehat H)$ is the maximum number of triangles in an $n$-vertex $\widehat H$-free graph. For fixed $k \ge 4$, Mubayi and Mukherjee conjectur...
An $n$-vertex graph $G$ is $(\rho, d)$-dense if for all $U \subseteq V(G)$ of size at least $\rho n$, the induced subgraph $G[U]$ contains at least $\frac{d}{2}|U|^2$ edges. In 2010, partly motivated by the theory of quasirandomness, Kohayakawa, Nagle, R\"{o}dl, and Schacht conjectured that, for any fixed graph $F$, th...
For a family $\mathcal{F}$ of $k$-graphs, $ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n\g...
We study edge complexity, the minimum Fourier $\ell^1/\ell^2$ ratio of a graph's adjacency matrix over cyclic vertex labelings. For $K_{m,n}$, with $1\leq m\leq n$, we reduce the problem to minimizing the Fourier $\ell^1$-norm of indicators of $m$-element subsets of $\mathbb Z_{m+n}$. We prove the sharp bound $\operato...
S. Ferguson, V. Gupta, A. Iosevich et al.· 0 citations
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