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Given a graph $F$, a graph $G$ is weakly $F$-saturated if all non-edges of $G$ can be added in some order so that each new edge introduces a copy of $F$. The weak saturation number $\operatorname{wsat}(n, F)$ is the minimum number of edges in a weakly $F$-saturated graph on $n$ vertices. Bollob\'as initiated the study of weak saturation in 1968 to study percolation processes, which originated in biology and have applications in physics and computer science. It was shown by Alon that for each $F$, there is a constant $w_F$ such that $\operatorname{wsat}(n, F) = w_Fn + o(n)$. We characterize all possible rational values of $w_F$, proving in particular that $w_F$ can equal any rational number at least $\frac 32$., Comment: 20 pages, 3 figures

In 1975, Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di asked for the smallest $\tau$ such that an $n \times n \times n$ tripartite graph with minimum degree $n + \tau$ must contain $K_{t, t, t}$, conjecturing that $\tau = \mathcal{O}(n^{1/2})$ for $t = 2$. We prove that $\tau = \mathcal{O}(n^{1 - 1/t})$ which confirms their conjecture and is best possible assuming the widely believed conjecture that $\operatorname{ex}(n, K_{t, t}) = \Theta(n^{2 - 1/t})$. Our proof uses a density increment argument. We also construct an infinite family of extremal graphs., Comment: 20 pages

Given a system $\mathcal{G} =\{G_1,G_2,\dots,G_m\}$ of graphs/digraphs/hypergraphs on the common vertex set $V$ of size $n$, an $m$-edge graph/digraph/hypergraph $H$ on $V$ is transversal in $\mathcal{G}$ if there exists a bijection $\phi :E(H)\rightarrow [m]$ such that $e \in E(G_{\phi(e)})$ for all $e\in E(H)$. In this survey, we consider extremal problems for transversal structures in graph systems. More precisely, we summarize some sufficient conditions that ensure the existence of transversal structures in graph/digraph/hypergraph systems, which generalize several classical theorems in extremal graph theory to transversal version. We also include a number of conjectures and open problems., Comment: 24 pages, 2 figures

We say that two vertices are twins if they have the same neighbourhood and that a graph is $K_r$-saturated if it does not contain $K_r$ but adding any new edge to it creates a $K_r$. In 1964, Erd\H{o}s, Hajnal and Moon showed that $sat(n,K_r)=(r-2)n+o(n)$ for $r \geq 3$, where $sat(n,K_r)$ is the minimum number of edges in a $K_r$-saturated graph on $n$ vertices, and determined the unique extremal graph. This graph has many twins, leading us to define $tsat(n,K_r)$ to be the minimum number of edges in a twin-free $K_r$-saturated graph on $n$ vertices. We show that $(5 +2/3)n + o(n) \leq tsat(n,K_3) \leq 6n+o(n)$ and that $\left(r+2\right)n + o(n) \leq tsat(n,K_r) \leq (r+3)n+o(n)$ for $r \geq 4$. We also consider a variant of this problem where we additionally require the graphs to have large minimum degree. Both of these problems turn out be intimately related to two other problems regarding maximally independent sets of a given size in $K_3$-free graphs and generalisations of these problems to $K_r$ with $r \geq 4$. The first problem is to maximise the number of maximally independent sets given the number of vertices and the second problem is to minimise the number of edges given the number of maximally independent sets. They are interesting in their own right., Comment: 25 pages,3 figures

For a family of graphs $\cal F$, a graph $G$ is $\cal F$-free if it does not contain a member of $\cal F$ as a subgraph. The Tur\'an number $\textrm{ex}(n,{\cal F})$ is the maximum number of edges in an $n$-vertex graph which is $\cal F$-free. Let ${\cal C}_{\geq k}$ be the set of cycles with length at least $k$. In this paper, we investigate the Tur\'an number of $\{{\cal C}_{\geq k}, F\}$ for a general graph $F$. To be precise, we determine $\textrm{ex}(n, \{{\cal C}_{\geq k}, F\})$ apart from a constant additive term, where $F$ either is a 2-connected nonbipartite graph or is a 2-connected bipartite graph under some conditions. This is an extension of a previous result on the Tur\'an number of $\{{\cal C}_{\geq k}, K_r\}$ by the first author, Ning, and the third author., Comment: corrected typos

The generalized Tur\'an number $\mathrm{ex}(n, H, \mathcal{F})$ is defined as the maximum number of copies of a graph $H$ in an $n$-vertex graph that does not contain any graph $F \in \mathcal{F}$. Alon and Frankl initiated the study of Tur\'an problems with a bounded matching number.In this paper, we establish stability results for generalized Tur\'an problems with bounded matching number.Using the stability results, we provide exact values of $\ex(n,K_r,\{F,M_{s+1}\})$ for $F$ being any non-bipartite graph or a path on $k$ vertices.

A graph $H$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to $G$ is denoted by $sat(G,F)$. Let $K_k^n$ be the complete $k$-partite graph containing $n$ vertices in each part and $C_\ell$ be the cycle of length $\ell$. In this paper we give an asymptotically tight bound of $sat(K_k^n,C_\ell)$ for all $ \ell \geq 4, k \geq 2$ except $(\ell,k)=(4,4)$. Moreover, we determined the exact value of $sat(K_k^n,C_\ell)$ for $ k>\ell=4 $ and $5 \geq \ell>k \geq 3$ and $(\ell,k)=(6,2)$., Comment: 31 pages, 21 figures, 9 theorems

In 2020, Coregliano and Razborov introduced a general framework to study limits of combinatorial objects, using logic and model theory. They introduced the abstract chromatic number and proved/reproved multiple Erd\H{o}s-Stone-Simonovits-type theorems in different settings. In 2022, Coregliano extended this by showing that similar results hold when we count copies of $K_t$ instead of edges. Our aim is threefold. First, we provide a purely combinatorial approach. Second, we extend their results by showing several other graph parameters and other settings where Erd\H{o}s-Stone-Simonovits-type theorems follow. Third, we go beyond determining asymptotics and obtain corresponding stability, supersaturation, and sometimes even exact results., Comment: 12 pages

The Wiener index of a graph is the sum of all the distances between any pair of vertices. We aim to describe graphs which minimize the Wiener index among all unicyclic graphs with fixed girth and given degree sequence. Depending on where the centroid of the graph is, we will present three candidates for the minimization, namely the greedy unicyclic graph, the cycle-centered graph and the out-greedy unicyclic graph., Comment: 24 pages

Let $\mathrm{ex}(n,H,\mathcal{F})$ be the maximum number of copies of $H$ in an $n$-vertex graph which contains no copy of a graph from $\mathcal{F}$. Thinking of $H$ and $\mathcal{F}$ as fixed, we study the asymptotics of $\mathrm{ex}(n,H,\mathcal{F})$ in $n$. We say that a rational number $r$ is \emph{realizable for $H$} if there exists a finite family $\mathcal{F}$ such that $\mathrm{ex}(n,H,\mathcal{F}) = \Theta(n^r)$. Using randomized algebraic constructions, Bukh and Conlon showed that every rational between $1$ and $2$ is realizable for $K_2$. We generalize their result to show that every rational between $1$ and $t$ is realizable for $K_t$, for all $t \geq 2$. We also determine the realizable rationals for stars and note the connection to a related Sidorenko-type supersaturation problem., Comment: 28 pages, 8 figures

A graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ but $H$ is a subgraph of $G + e$ for any edge $e$ in $\overline{G}$. The saturation number $sat(n,H)$ for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. The $sat(n, K_{p_1} \cup K_{p_2} \cup K_{p_3})$ with $p_3 \ge p_1 + p_2$ was given in [Discrete Math. 347 (2024) 113868]. In this paper, $sat(n,K_{p_1} \cup K_{p_2} \cup K_{p_3} \cup K_{p_4})$ with $p_{i+1} - p_i \ge p_1$ for $2 \le i\le 3$ and $4\le p_1\le p_2$ is determined., Comment: 17pages

Let $H$ be a fixed graph, a graph G is $H$-saturated if it has no copy of $H$ in $G$, but the addition of any edge in $E(\overline G)$ to $G$ results in an $H$-subgraph. The saturation number sat$(n,H)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. In this paper, we determine the saturation number sat$(n,P_7+tP_2)$ for $n\geq \frac {14}{5}t+27$ and characterize the extremal graphs for $n\geq \frac{14}{13}(3t+25)$., Comment: 11 pages

Given a collection $\mathcal{G} =\{G_1,G_2,\dots,G_m\}$ of graphs on the common vertex set $V$ of size $n$, an $m$-edge graph $H$ on the same vertex set $V$ is transversal in $\mathcal{G}$ if there exists a bijection $\varphi :E(H)\rightarrow [m]$ such that $e \in E(G_{\varphi(e)})$ for all $e\in E(H)$. Denote $\delta(\mathcal{G}):=\operatorname*{min}\left\{\delta(G_i): i\in [m]\right\}$. In this paper, we first establish a minimum degree condition for the existence of transversal Hamilton paths in $\mathcal{G}$: if $n=m+1$ and $\delta(\mathcal{G})\geq \frac{n-1}{2}$, then $\mathcal{G}$ contains a transversal Hamilton path. This solves a problem proposed by [Li, Li and Li, J. Graph Theory, 2023]. As a continuation of the transversal version of Dirac's theorem [Joos and Kim, Bull. Lond. Math. Soc., 2020] and the stability result for transversal Hamilton cycles [Cheng and Staden, arXiv:2403.09913v1], our second result characterizes all graph collections with minimum degree at least $\frac{n}{2}-1$ and without transversal Hamilton cycles. We obtain an analogous result for transversal Hamilton paths. The proof is a combination of the stability result for transversal Hamilton paths or cycles, transversal blow-up lemma, along with some structural analysis., Comment: 33 pages, 10 figures

Let $H$ be a fixed graph. A graph $G$ is called {\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the minimum number of edges over all $H$-saturated graphs of order $n$, and $Sat(n,H)$ denote the family of $H$-saturated graphs with $sat(n,H)$ edges and $n$ vertices. In this paper, we resolve a conjecture of Chen and Yuan in[Discrete Math. 347(2024)113868] by determining $Sat(n,K_p\cup (t-1)K_q)$ for every $2\le p\le q$ and $t\ge 2$., Comment: 7 pages, 2 figures

Given a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n, \mathcal{F})$ denotes the maximum number of edges in any $\mathcal{F}$-free graph on $n$ vertices. Recently, Alon and Frankl studied of maximum number of edges in an $n$-vertex $\{K_{k+1}, M_{s+1}\}$-free graph, where $K_{k+1}$ is a complete graph on $k+1$ vertices and $M_{s+1}$ is a matching of $s+1$ edges. They determined the exact value of $ex(n, \{K_{k+1},M_{s+1}\})$. In this paper, we extend the matching $M_{s+1}$ to star forest $(s+1)S_l$, and determine the exact value of $ex(n, \{K_{k+1},(s+1)S_l\})$ for sufficiently large enough $n$. Furthermore, all the extremal graphs are obtained.

We estimate the maximum possible number of cliques of size $r$ in an $n$-vertex graph free of a fixed complete $r$-partite graph $K_{s_1, s_2, \ldots, s_r}$. By viewing every $r$-clique as a hyperedge, the upper bound on the Tur\'an number of the complete $r$-partite hypergraphs gives the upper bound $O\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. We improve this to $o\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. The main tool in our proof is the graph removal lemma. We also provide several lower bound constructions., Comment: 6 pages

The saturation number $\operatorname{sat}(n, H)$ of a graph $H$ and positive integer $n$ is the minimum size of an $n$-vertex graph which does not contain a subgraph isomorphic to $H$ but to which the addition of any edge creates such a subgraph. Erd\H{o}s, Hajnal, and Moon first studied saturation numbers of complete graphs, and Cameron and Puleo introduced a general lower bound on $\operatorname{sat}(n,H)$. In this paper, we present another lower bound on $\operatorname{sat}(n, H)$ with a strengthening when $H$ is triangle-free. Demonstrating its effectiveness, we determine the saturation numbers of diameter-$3$ trees up to an additive constant; these are double stars $S_{s,t}$ on $s + t$ vertices whose centers have degrees $s$ and $t$. Faudree, Faudree, Gould, and Jacobson determined that $\operatorname{sat}(n, S_{t,t}) = (t-1)n/2 + O(1)$. We show that $\operatorname{sat}(n,S_{s,t}) = (st+s)n/(2t+4) + O(1)$ when $s < t$. We also determine lower and upper bounds on the saturation numbers of certain diameter-$4$ caterpillars.

The independent domination number of a finite graph G is the minimum cardinality of an independent dominating set of vertices. The independent bondage number of G is the minimum cardinality of a set of edges whose deletion results in a graph with a larger independent domination number than that of G. In this research, we enhance the existing upper bound on the independent bondage number of a planar graph with a minimum degree of at least three by identifying specific configurations within such planar graphs.

In this paper, we explore the concept of total bondage in finite graphs without isolated vertices. A vertex set $D$ is considered a total dominating set if every vertex $v$ in the graph $G$ has a neighbor in $D$. The minimum cardinality of all total dominating sets in $G$ is denoted as $\gamma_t(G)$. A total bondage edge set $B$ is a subset of the edges of $G$ such that the removal of $B$ from $G$ does not create isolated vertices, and the total dominating number of the resulting graph $G-B$ is strictly greater than $\gamma_t(G)$. The total bondage number of $G$, denoted $b_t(G)$, is defined as the minimum cardinality of such total bondage edge sets. Our paper establishes upper bounds on $b_t(G)$ based on the maximum degree of a graph. Notably, for planar graphs with minimum degree $\delta(G) \geq 3$, we prove $b_t(G) \leq \Delta + 8$ or $b_t(G) \leq 10$. Additionally, for a connected planar graph with $\delta(G) \geq 3$ and $g(G) \geq 4$, we show that $b_t(G) \leq \Delta + 3$ if $G$ does not contain an edge with degree sum at most 7. We also improve some upper bounds of the total bondage number for trees, enhance existing lemmas, and find upper bounds for total bondage in specific graph classes., Comment: 11 pages

The classical extremal function for a graph $H$, $ex(K_n, H)$ is the largest number of edges in a subgraph of $K_n$ that contains no subgraph isomorphic to $H$. Note that defining $ex(K_n, H-ind)$ by forbidding induced subgraphs isomorphic to $H$ is not very meaningful for a non-complete $H$ since one can avoid it by considering a clique. For graphs $F$ and $H$, let $ex(K_n, \{F, H-ind\})$ be the largest number of edges in an $n$-vertex graph that contains no subgraph isomorphic to $F$ and no induced subgraph isomorphic to $H$. Determining this function asymptotically reduces to finding either $ex(K_n, F)$ or $ex(K_n, H)$, unless $H$ is a biclique or both $F$ and $H$ are bipartite. Here, we consider the bipartite setting, $ex(K_{n,n}, \{F, H-ind\})$ when $K_n$ is replaced with $K_{n,n}$, $F$ is a biclique, and $H$ is a bipartite graph. Our main result, a strengthening of a result by Sudakov and Tomon, implies that for any $d\geq 2$ and any $K_{d,d}$-free bipartite graph $H$ with each vertex in one part of degree either at most $d$ or a full degree, so that there are at most $d-2$ full degree vertices in that part, one has $ex(K_{n,n}, \{K_{t,t}, H-ind\}) = o(n^{2-1/d})$. This provides an upper bound on the induced Tur\'an number for a wide class of bipartite graphs and implies in particular an extremal result for bipartite graphs of bounded VC-dimension by Janzer and Pohoata., Comment: New references are added

For a family $\mathcal{F}$ of graphs, $sat(n,\mathcal{F})$ is the minimum number of edges in a graph $G$ on $n$ vertices which does not contain any of the graphs in $\mathcal{F}$ but such that adding any new edge to $G$ creates a graph in $\mathcal{F}$. For singleton families $\mathcal{F}$, Tuza conjectured that $sat(n,\mathcal{F})/n$ converges and Truszczynski and Tuza discovered that either $sat(n,\mathcal{F})= \left(1-1/r\right)n+o(n)$ for some integer $r \geq 1$ or $ sat(n,\mathcal{F}) \geq n+o(n) $. This is often cited in the literature as the main progress towards proving Tuza's Conjecture. Unfortunately, the proof is flawed. We give a correct proof, which requires a novel construction. Moreover, for finite families $\mathcal{F}$, we completely determine the possible asymptotic behaviours of $sat(n,\mathcal{F})$ in the sparse regime $sat(n,\mathcal{F}) \leq n+o(n)$. Finally, we essentially determine which sequences of integers are of the form $\left(sat(n,\mathcal{F})\right)_{n \geq 0}$ for some (possibly infinite) family $\mathcal{F}$., Comment: 22 pages, 15 figures

In this paper, we prove that any simple $\{C_3,C_5\}$-free non-empty connected graph $G$ with LLY curvature bounded below by $\kappa>0$ has the order at most $2^{\frac{2}{\kappa}}$. This upper bound is achieved if and only if $G$ is a hypercube $Q_d$ and $\kappa=\frac{2}{d}$ for some integer $d\geq 1$., Comment: 10 pages

Many well-studied problems in extremal combinatorics deal with the maximum possible size of a family of objects in which every pair of objects satisfies a given restriction. One problem of this type was recently raised by Alon, Gujgiczer, K\"orner, Milojevi\'c and Simonyi. They asked to determine the maximum size of a family $\mathcal{G}$ of graphs on $[n]$, such that for every two $G_1,G_2 \in \mathcal{G}$, the graphs $G_1 \setminus G_2$ and $G_2 \setminus G_1$ are isomorphic. We completely resolve this problem by showing that this maximum is exactly $2^{\frac{1}{2}\big(\binom{n}{2} - \lfloor \frac{n}{2}\rfloor\big)}$ and characterizing all the extremal constructions. We also prove an analogous result for $r$-uniform hypergraphs.

An {\it inversion} of a tournament $T$ is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let ${\rm inv}_k(T)$ be the minimum length of a sequence of inversions using sets of size at most $k$ that result in the transitive tournament. Let ${\rm inv}_k(n)$ be the maximum of ${\rm inv}_k(T)$ taken over $n$-vertex tournaments. It is well-known that ${\rm inv}_2(n)=(1+o(1))n^2/4$ and it was recently proved by Alon et al. that ${\rm inv}(n):={\rm inv}_{n}(n)=n(1+o(1))$. In these two extreme cases ($k=2$ and $k=n$), random tournaments are asymptotically extremal objects. It is proved that the random tournament {\em does not} asymptotically attain ${\rm inv}_k(n)$ when $k \ge k_0$ and conjectured that ${\rm inv}_3(n)$ is (only) attained by (quasi) random tournaments. It is further proved that $(1+o(1)){\rm inv}_3(n)/n^2 \in [\frac{1}{12}, 0.0992)$ and $(1+o(1)){\rm inv}_k(n)/n^2 \in [\frac{1}{2k(k-1)}+\delta_k, \frac{1}{2 \lfloor k^2/2 \rfloor}-\epsilon_k]$ where $\epsilon_k > 0$ for all $k \ge 3$ and $\delta_k > 0$ for all $k \ge k_0$.

For a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. The maximum number of edges in an $n$-vertex connected graph containing no member of $\mathcal{F}$ as a subgraph is denoted by $ex_{conn}(n,\mathcal{F})$. Let $P_k$ be the path on $k$ vertices and $H$ be a graph with chromatic number more than $2$. Katona and Xiao [Extremal graphs without long paths and large cliques, European J. Combin., 2023 103807] posed the following conjecture: Suppose that the chromatic number of $H$ is more than $2$. Then $ex\big(n,\{H,P_k\}\big)=n\max\big\{\big\lfloor \frac{k}{2}\big\rfloor-1,\frac{ex(k-1,H)}{k-1}\big\}+O_k(1)$. In this paper, we determine the exact value of $ex_{conn}\big(n,\{P_k,H\}\big)$ for sufficiently large $n$. Moreover, we obtain asymptotical result for $ex\big(n,\{P_k,H\}\big)$, which solves the conjecture proposed by Katona and Xiao., Comment: 16 pages, 6 conferences

For given positive integers $r\ge 3$, $n$ and $e\le \binom{n}{2}$, the famous Erd\H os--Rademacher problem asks for the minimum number of $r$-cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lov\'asz and Simonovits from the 1970s states that, for every $r\ge 3$, if $n$ is sufficiently large then, for every $e\le \binom{n}{2}$, at least one extremal graph can be obtained from a complete partite graph by adding a triangle-free graph into one part. In this note, we explicitly write the minimum number of $r$-cliques predicted by the above conjecture. Also, we describe what we believe to be the set of extremal graphs for any $r\ge 4$ and all large~$n$, amending the previous conjecture of Pikhurko and Razborov., Comment: revised according to referee's suggestions

The classical Zarankiewicz's problem asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{t,t}$. In one of the cornerstones of extremal graph theory, K\H{o}v\'ari S\'os and Tur\'an proved an upper bound of $O(n^{2-\frac{1}{t}})$. In a celebrated result, Fox et al. obtained an improved bound of $O(n^{2-\frac{1}{d}})$ for graphs of VC-dimension $d$ (where $d

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an $\mathcal{H}$-universal graph can have. With the aim of unifying a number of recent results, we consider a family of graphs with bounded density. In particular, we construct a graph with $O_d\left( n^{2 - 1/(\lceil d \rceil + 1)} \right)$ edges which contains every $n$-vertex graph with density at most $d \in \mathbb{Q}$ ($d \ge 1$), which is close to a lower bound $\Omega(n^{2 - 1/d - o(1)})$ obtained by counting lifts of a carefully chosen (small) graph. When restricting the maximum degree of such graphs to be constant, we obtain a near-optimal universality. If we further assume $d \in \mathbb{N}$, we get an asymptotically optimal construction., Comment: 14 pages, updated version focusing on density, with new title and additional author

The systematic study of Tur\'an-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. in 2020. Here we characterize connected edge-ordered graphs with linear extremal functions and show that the extremal function of other connected edge-ordered graphs is $\Omega(n\log n)$. This characterization and dichotomy are similar in spirit to results of F\"uredi et al. (2020) about vertex-ordered and convex geometric graphs. We also extend the study of extremal function of short edge-ordered paths by Gerbner et al. to some longer paths., Comment: 20 pages, 5 figures, 1 table