Related papers: Bipartite decomposition of random graphs

For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \geq b_r(G)$ for any $G$, and Tur\'an's Theorem says that equality holds for complete graphs.…

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a…

A set of edges $T$ in a graph $G$ is triangle-independent if $T$ contains at most one edge from each triangle in $G$. Let $\alpha_1(G)$ denote the maximum size of the triangle-independent set in $G$, and let $\tau_B(G)$ denote minimum size…

For a graph $G$, let $\tau(G)$ be the maximum number of colors such that there exists an edge-coloring of $G$ with no two color classes being isomorphic. We investigate the behavior of $\tau(G)$ when $G=G(n, p)$ is the classical…

A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t,t}$ in the bipartite complement of $G$. Given an $n \times n$ bipartite graph $G$, let $\beta(G)$ be the largest $k$ for which $G$ has a bi-hole of size $k$. We prove that \[…

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D \geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

Let $c$ denote the largest constant such that every $C_{6}$-free graph $G$ contains a bipartite and $C_4$-free subgraph having $c$ fraction of edges of $G$. Gy\H{o}ri et al. showed that $\frac{3}{8} \le c \le \frac{2}{5}$. We prove that…

For a graph $G$, denote by $t(G)$ (resp. $b(G)$) the maximum size of a triangle-free (resp. bipartite) subgraph of $G$. Of course $t(G) \geq b(G)$ for any $G$, and a classic result of Mantel from 1907 (the first case of Tur\'an's Theorem)…

A classical result in extremal graph theory is Mantel's Theorem, which states that every maximum triangle-free subgraph of $K_n$ is bipartite. A sparse version of Mantel's Theorem is that, for sufficiently large $p$, every maximum…

For a graph $G$ with order $2n$ and a perfect matching, let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ respectively. Then $0\leq f(G)\leq F(G)\leq n-1$. Liu and Zhang [10] ever proposed a conjecture: $e(G)\geq…

Let $G_{n,p}$ be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let $G_{n,n,p}$ be the random bipartite graph on $n+n$ vertices, where each $e\in [n]^2$ appears as an edge independently with probability $p$. For a graph $G=(V,E)$,…

We analyse the eigenvalues of Erd\"os--R\'enyi random bipartite graphs. In particular, we consider $p$ satisfying $n_{1}p=\Omega(\sqrt{n_{1}p}\log^{3}(n_{1})),$ $n_{2}p=\Omega(\sqrt{n_{2}p}\log^{3}(n_{2})),$ and let $G\sim…

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Engbers and Galvin asked how large $i_t(G)$ could be in graphs with minimum degree at least $\delta$. They further conjectured that when $n\geq 2\delta$ and $t\geq…

In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which…

For a graph $G$ and a hereditary property $\mathcal{P}$, let $\text{ex}(G,\mathcal{P})$ denote the maximum number of edges of a subgraph of $G$ that belongs to $\mathcal{P}$. We prove that for every non-trivial hereditary property…

A sparse version of Mantel's Theorem is that, for sufficiently large $p$, with high probability (w.h.p.), every maximum triangle-free subgraph of $G(n,p)$ is bipartite. DeMarco and Kahn proved this for $p>K \sqrt{\log n/n}$ for some…

Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $\gamma_{t}$($G$), is the minimum cardinality of a total…

Tuza conjectured that for every graph $G$, the maximum size $\nu$ of a set of edge-disjoint triangles and minimum size $\tau$ of a set of edges meeting all triangles satisfy $\tau \leq 2\nu$. We consider an edge-weighted version of this…

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erd\H{o}s-Gallai Theorem in random graphs. In particular, we determine, up to a constant…

Given a graph $F$, the random Tur\'an problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete…