Ya-Chen Chen
A graph is K_{2,3}-saturated if it has no subgraph isomorphic to K_{2,3}, but does contain a K_{2,3} after the addition of any new edge. We prove that the minimum number of edges in a K_{2,3}-saturated graph on n >= 5 vertices is sat(n, K_{2,3}) = 2n - 3.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Minimum Weakly Saturated Graphs and Bootstrap Percolation in General Host Graphs
本文研究最小弱饱和图类及其与H-自举渗流的关联，探讨一般宿主图下的弱饱和问题，明确了弱饱和数与最小弱饱和图的定义。
A graph $G$ is $H$-saturated if it contains no copy of $H$ as a subgraph but the addition of any new edge to $G$ creates a copy of $H$. In this paper we are interested in the function sat$_{t}(n,p)$, defined to be the minimum number of edges that a $K_{p}$-saturated graph on $n$ vertices can have if it has minimum degree at least $t$. We prove that sat$_{t}(n,p) = tn - O(1)$, where the limit is taken as $n$ tends to infinity. This confirms a conjecture of Bollobás when $p = 3$. We also present constructions for graphs that give new upper bounds for sat$_{t}(n,p)$ and discuss an analogous probl…
For $n \geq 15$, we prove that the minimum number of triangles in an $n$-vertex $K_4$-saturated graph with minimum degree 4 is exactly $2n-4$, and that there is a unique extremal graph. This is a triangle version of a result of Alon, Erdős, Holzman, and Krivelevich from 1996. Additionally, we show that for any $s > r \geq 3$ and $t \geq 2 (s-2)+1$, there is a $K_s$-saturated $n$-vertex graph with minimum degree $t$ that has $\binom{ s-2}{r-1}2^{r-1} n + c_{s,r,t}$ copies of $K_r$. This shows that unlike the number of edges, the number of $K_r$'s ($r >2$) in a $K_s$-saturated graph is not force…
A graph $G$ is called $H$-saturated if it does not contain any copy of $H$, but for any edge $e$ in the complement of $G$ the graph $G+e$ contains some $H$. The minimum size of an $n$-vertex $H$-saturated graph is denoted by $\sat(n,H)$. We prove $$\sat(n,C_k) = n + n/k + O((n/k^2) + k^2)$$ holds for all $n\geq k\geq 3$, where $C_k$ is a cycle with length $k$. We have a similar result for semi-saturated graphs $$\ssat(n,C_k) = n + n/(2k) + O((n/k^2) + k).$$ We conjecture that our three constructions are optimal.
Motivated by the work of Anstee, Griggs, and Sali on forbidden submatrices and the extremal sat-function for graphs, we introduce sat-type problems for matrices. Let F be a family of k-row matrices. A matrix M is called F-admissible if M contains no submatrix G\in F (as a row and column permutation of G). A matrix M without repeated columns is F-saturated if M is F-admissible but the addition of any column not present in M violates this property. In this paper we consider the function sat(n,F) which is the minimum number of columns of an F-saturated matrix with n rows. We establish the estimat…
An $n$-by-$n$ bipartite graph is $H$-saturated if the addition of any missing edge between its two parts creates a new copy of $H$. In 1964, Erdős, Hajnal and Moon made a conjecture on the minimum number of edges in a $K_{s,s}$-saturated bipartite graph. This conjecture was proved independently by Wessel and Bollobás in a more general, but ordered, setting: they showed that the minimum number of edges in a $K_{(s,t)}$-saturated bipartite graph is $n^2-(n-s+1)(n-t+1)$, where $K_{(s,t)}$ is the "ordered" complete bipartite graph with $s$ vertices in the first color class and $t$ vertices in the …
Minimum saturated graphs for unions of cliques
该研究解决了Chen与Yuan提出的猜想，确定了团的并图$K_p\cup (t-1)K_q$对应的$n$阶最小饱和图族$Sat(n,K_p\cup (t-1)K_q)$，完善了图论中饱和数相关的结论。
The minimum spectral radius of $tP_4$-saturated graphs
本文研究了$tP_4$-饱和图的谱半径问题，证明了其最小谱半径下界，并刻画了等号成立的图结构，同时指出特定条件下最小化谱半径与边数的集合不相交。
Given a graph $F$, a graph $G$ is {\it uniquely $F$-saturated} if $F$ is not a subgraph of $G$ and adding any edge of the complement to $G$ completes exactly one copy of $F$. In this paper we study uniquely $C_t$-saturated graphs. We prove the following: (1) a graph is uniquely $C_5$-saturated if and only if it is a friendship graph. (2) There are no uniquely $C_6$-saturated graphs or uniquely $C_7$-saturated graphs. (3) For $t\ge6$, there are only finitely many uniquely $C_t$-saturated graphs (we conjecture that in fact there are none).
Geschke proved that there is clopen graph on $2^ω$ which is 3-saturated, but the clopen graphs on $2^ω$ do not even have infinite subgraphs that are 4-saturated; however there is $F_σ$ graph that is $ω_1$-saturated. It turns out that there is no closed graph on $2^ω$ which is $ω$-saturated. In this note we complete this picture by proving that for every $n$ there is an $n$-saturated closed graph on the Cantor space $2^ω$. The key lemma is based on probabilistic argument. The final construction is an inverse limit of finite graphs.
The Minimum Number of Edges in $(p+1)K_2$-Saturated Graphs
本文针对极值图论中确定$(p+1)K_2$-饱和图最小边数的公开问题，通过细化结构分析推导边数参数公式，完整解决该问题并推广已有结论，同时还原经典边数上界。
Minimum saturated graphs without $4$-cycles and $5$-cycles
本文研究不含4-环与5-环的图的饱和数问题，确定了n个顶点的{C₄,C₅}-饱和图的最小边数公式，推进了环族饱和数的精确计算研究。
All minimum $C_4$-saturated multipartite graphs
本文确定了完全 $k$-部图 $K_k^n$ 中 $C_4$-饱和子图的最小边数 $sat(K_4^n,C_4)$，并刻画了所有 $n\ge 2$ 和 $k\ge 4$ 时的极值结构。
A detour of a graph G is a longest path in G. The detour order of G is the number of vertices in a detour of G. A graph is said to be detour-saturated if the addition of any edge increases strictly the detour order. L.W. Beineke, J.E. Dunbar and M. Frick asked the following three questions in 2005. (1) What is the smallest order of a detour-saturated graph of girth 4? (2) Let Pr be the graph obtained from the Petersen graph by splitting one of its vertices into three leaves. Is Pr the smallest triangle-free detour-saturated graph? (3) Does there exist a detour-saturated graph with finite girth…
In a recent paper, Gerbner, Patkós, Tuza and Vizer studied regular $F$-saturated graphs. One of the essential questions is given $F$, for which $n$ does a regular $n$-vertex $F$-saturated graph exist. They proved that for all sufficiently large $n$, there is a regular $K_3$-saturated graph with $n$ vertices. We extend this result to both $K_4$ and $K_5$ and prove some partial results for larger complete graphs. Using a variation of sum-free sets from additive combinatorics, we prove that for all $k \geq 2$, there is a regular $C_{2k+1}$-saturated with $n$ vertices for infinitely many $n$. Stud…
We call a family $\mathcal{F}$ of subsets of $[n]$ $s$-saturated if it contains no $s$ pairwise disjoint sets, and moreover no set can be added to $\mathcal{F}$ while preserving this property (here $[n] = \{1,\ldots,n\}$). More than 40 years ago, Erdős and Kleitman conjectured that an $s$-saturated family of subsets of $[n]$ has size at least $(1 - 2^{-(s-1)})2^n$. It is easy to show that every $s$-saturated family has size at least $\frac{1}{2}\cdot 2^n$, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of $(1/2 + \varepsilon)2^n$, for some fixed $\varepsi…
For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $n\to\infty$.
Let $P_{n}$ be a path graph on $n$ vertices. We say that a graph $G$ is $P_{n}$-induced-saturated if $G$ contains no induced copy of $P_{n}$, but deleting any edge of $G$ as well as adding to $G$ any edge of $G^{c}$ creates such a copy. Martin and Smith (2012) showed that there is no $P_{4}$-induced-saturated graph. On the other hand, there trivially exist $P_{n}$-induced-saturated graphs for $n=2,3$. Axenovich and Csikós (2019) ask for which integers $n \geq 5$ do there exist $P_{n}$-induced-saturated graphs. Räty (2019) constructed such a graph for $n=6$, and Cho, Choi and Park (2019) later …
Induced-saturated graphs exist for even cycles
该研究解决了环的诱导饱和图存在性问题，证明除长度为3的环外，所有环都存在对应的诱导饱和图，对q≥3构造了C_{2q+2}-诱导饱和图。
A graph $G$ is $F$-saturated if it contains no copy of $F$ as a subgraph but the addition of any new edge to $G$ creates a copy of $F$. We prove that for $s \geq 3$ and $t \geq 2$, the minimum number of copies of $K_{1,t}$ in a $K_s$-saturated graph is $Θ( n^{t/2})$. More precise results are obtained when $t = 2$ where the problem is related to Moore graphs with diameter 2 and girth 5. We prove that for $s \geq 4$ and $t \geq 3$, the minimum number of copies of $K_{2,t}$ in an $n$-vertex $K_s$-saturated graph is at least $Ω( n^{t/5 + 8/5})$ and at most $O(n^{t/2 + 3/2})$. These results answer …
A matroid is supersolvable if it has a maximal chain of flats each of which is modular. A matroid is saturated if every round flat is modular. In this article we present supersolvable saturated matroids as analogues to chordal graphs, and we show that several results for chordal graphs hold in this matroid context. In particular, we consider matroid analogues of the reduced clique graph and clique trees for chordal graphs. The latter is a maximum-weight spanning tree of the former. We also show that the matroid analogue of a clique tree is an optimal decomposition for the matroid parameter of …
For a graph $H$, a graph $G$ is $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but either removing an edge from $G$ or adding a non-edge to $G$ creates an induced copy of $H$. Depending on the graph $H$, an $H$-induced-saturated graph does not necessarily exist. In fact, Martin and Smith (2012) showed that $P_4$-induced-saturated graphs do not exist, where $P_k$ denotes a path on $k$ vertices. Axenovich and Csikós (2019) asked the existence of $P_k$-induced-saturated graphs for $k \ge 5$; it is easy to construct such graphs when $k\in\{2, 3\}$. Recently, Räty constructed…
Some results on minimum saturated graphs
本文研究了极小饱和图问题，确定了 $\sat(n,\{K_3,P_k\})$ 的精确值，推导了 $\sat(n,K_3\cup P_k)$ 的两个界，并给出了 $\sat(n,K_1\lor F)$ 的精确值。
Given graphs $G$ and $H$, $G$ is $H$-saturated if $G$ does not contain a copy of $H$ but the addition of any edge $e\notin E(G)$ creates at least one copy of $H$ within $G$. The edge spectrum of $H$ is the set of all possible sizes of an $H$-saturated graph on $n$ vertices. Let $K_4^-$ be a graph obtained from $K_4$ by deleting an edge. In this note, we show that (a) if $G$ is a $K_4^-$-saturated graph with $|V(G)|=n$ and $|E(G)|>\lfloor \frac{n-1}{2} \rfloor \lceil \frac{n-1}{2} \rceil +2$, then $G$ must be a bipartite graph; (b) there exists a $K_4^-$-saturated non-bipartite graph on $n\ge 1…
A graph $G$ is $H$-saturated for a graph $H$, if $G$ does not contain a copy of $H$ but adding any new edge to $G$ results in such a copy. An $H$-saturated graph on a given number of vertices always exists and the properties of such graphs, for example their highest density, have been studied intensively. A graph $G$ is $H$-induced-saturated if $G$ does not have an induced subgraph isomorphic to $H$, but adding an edge to $G$ from its complement or deleting an edge from $G$ results in an induced copy of $H$. It is not immediate anymore that $H$-induced-saturated graphs exist. In fact, Martin a…
For a fixed graph $F$, a graph $G$ is said to be $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$ but does contain $F$ after the addition of any new edge. Let $M_k$ be a matching consisting of $k$ edges and $S_{n,k}$ be the join graph of a complete graph $K_k$ and an empty graph $\overline{K_{n-k}}$. In this paper, we prove that for $s \geq3$ and $k\geq 2$, $S_{n,s-2}$ contains the minimum number of $M_k$ among all $n$-vertex $K_s$-saturated graphs for sufficiently large $n$, and when $k \leq s-2$, it is the unique extremal graph. In addition, we also show that $S_{n,1}$ is t…
The saturation number of a graph $F$, written $\textup{sat}(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. One of the earliest results on saturation numbers is due to Erdős, Hajnal, and Moon who determined $\textup{sat}(n,K_r)$ for all $r \geq 3$. Since then, saturation numbers of various graphs and hypergraphs have been studied. Motivated by Alon and Shikhelman's generalized Turán function, Kritschgau et.\ al.\ defined $\textup{sat}(n,H,F)$ to be the minimum number of copies of $H$ in an $n$-vertex $F$-saturated graph. They proved, among other things, that $\textu…
Infinite induced-saturated graphs
针对不存在有限诱导饱和图的有限图（如 $P_4$），本文证明每个非团、非独立集的有限图 $H$ 都存在可数的 $H$-诱导饱和图，并给出对局部有限更改也饱和的更强构造。
Uniquely $C_{4}^{+}$-saturated graphs
本文研究了唯一 $C_{4}^{+}$-饱和图，证明了其围长只能是 3 或 4，给出了围长为 4 时的强正则图参数条件，并得出了关于三角形数量与顶点数的不存在性结论。
Gerbner, Patkós, Tuza, and Vizer recently initiated the study of $F$-saturated regular graphs. One of the essential problems in this line of research is determining when such a graph exists. Using generalized sum-free sets we prove that for any odd integer $k \geq 5$, there is an $n$-vertex regular $C_k$-saturated graph for all $n \geq n_k$. Our proof is based on constructing a special type of sum-free set in $\mathbb{Z}_n$. We prove that for all even $\ell \geq 4$ and integers $n > 12 \ell^2 + 36 \ell + 24$, there is a symmetric complete $( \ell , 1)$-sum-free set in $\mathbb{Z}_n$. We pose t…
The problem of determining the Turán number of $C_4$ is a well studied problem that dates back to a paper of Erdös from 1938. It is known that Sidon sets can be used to construct $C_4$-free graphs. If $\A$ is a Sidon set in the abelian group $X$, the sum graph $G_{X, \A}$ with vertex set $X$ and edges set $E=\{\{x, y\}:x\neq y, x+y\in \A\}$ is $C_4$-free. Using the sum graph of a Sidon set of type Singer we verify a conjecture of Erdös and Simonovits concerning the number of copies of $C_4$ in a graph with $ex(q^2+q+1, C_4)+1$ edges. Further, we give a sufficient condition for the sum graph of…
A graph $H^{\prime}$ is $(H, G)$-saturated if it is $G$-free and the addition of any edge of $H$ not in $H^{\prime}$ creates a copy of $G$. The saturation number $sat(H, G)$ is the minimum number of edges in a $(H, G)$-saturated graph. We investigate bounds on the saturation number of trees $T$ in the $n$-dimensional hypercube $Q_n$. We first present a general lower bound on the saturation number based on the minimum degree of non-leaves. From there, we suggest two general methods for constructing $T$-saturated subgraphs of $Q_n$, and prove nontrivial upper bounds for specific types of trees, …
This paper considers an edge minimization problem in saturated bipartite graphs. An $n$ by $n$ bipartite graph $G$ is $H$-saturated if $G$ does not contain a subgraph isomorphic to $H$ but adding any missing edge to $G$ creates a copy of $H$. More than half a century ago, Wessel and Bollobás independently solved the problem of minimizing the number of edges in $K_{(s,t)}$-saturated graphs, where $K_{(s,t)}$ is the `ordered' complete bipartite graph with $s$ vertices from the first color class and $t$ from the second. However, the very natural `unordered' analogue of this problem was considered…
A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with $n$ vertices and $O(n)$ edges. For every $k>1$, we give similar constructions for $k$-simple topological graphs, that is, for graphs drawn in the plane so that any two edges have at most $k$ points in common. We show that in any $k$-simple topological graph, any two independent vertic…
This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy of $F$. We first resolve a fundamental question of minimizing the number of cliques of size $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait, and Timmons. We also go further and prove a corresponding stability result. Next we minimize the number of cycles of length $r$ in a $K…
A graph is $F$-saturated if it is $F$-free but the addition of any edge creates a copy of $F$. In this paper we study the quantity $\mathrm{sat}(n, H, F)$ which denotes the minimum number of copies of $H$ that an $F$-saturated graph on $n$ vertices may contain. This parameter is a natural saturation analogue of Alon and Shikhelman's generalized Turán problem, and letting $H = K_2$ recovers the well-studied saturation function. We provide a first investigation into this general function focusing on the cases where the host graph is either $K_s$ or $C_k$-saturated. Some representative interestin…
Write $ρ\left( G\right) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}\vee\overline{K}_{n-r}.$ Let $n>r\geq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined exactly the minimum value of $ρ\left( G\right) $ for $r=2$, and found an asymptotically tight bound on $ρ\left( G\right) $ for $r\geq3.$ They also conjectured that \[ ρ\left( G\right) >ρ\left( S_{n,r-1}\right) , \] unless $G=S_{n,r-1}.$ In this note their conjecture is proved.
On rainbow saturated graphs with minimum number of edges
本文证明彩虹饱和数二分法：含孤立边的图饱和数为常数，否则为线性；并确定广义友谊图的渐近精确值。
Saturated $2$-planar drawings with few edges
该研究针对饱和2-平面图的最小边数问题，分别在两条边最多交叉一次和可多次交叉的条件下，推导得出n-1的下界和⌊2n/3⌋的紧界，补充了k>0时饱和k-平面图边数的相关结论。
We study the ratio, in a finite graph, of the sizes of the largest matching in any pair of disjoint matchings with the maximum total number of edges and the largest possible matching. Previously, it was shown that this ratio is between 4/5 and 1, and the class of graphs achieving 4/5 was completely characterized. In this paper, we first show that graph decompositions into paths and even cycles provide a new way to study this ratio. We then use this technique to characterize the graphs achieving ratio 1 among all graphs that can be covered by a certain choice of a maximum matching and maximum d…
Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$ but the addition of any edge $e\notin E(G)$ creates at least one copy of some $F\in\mathcal{F}$ within $G$. The minimum size of an $\mathcal{F}$-saturated graph on $n$ vertices are called the saturation number, denoted by $\sat(n, \mathcal{F})$. Let $\mathcal{C}_{\ge r}$ be the family of cycles of length at least $r$. Ferrara et al. (2012) gave lower and upper bounds of $\sat(n, C_{\ge r})$ and determined the exact values of $\s…
For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the plane, a drawing $D \in \mathcal{D}$ is \emph{saturated} when the addition of any edge to $D$ results in $D' \notin \mathcal{D}$ - this is analogous to saturated graphs in a graph class as introduced by Turán (1941) and Erdős, Hajnal, and Moon (1964). We focus on $k$-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most $k$ times, and the classes $\mathcal{D}$ of all $k$-planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing mo…
An $\mathcal{F}$-saturated $r$-graph is a maximal $r$-graph not containing any member of $\mathcal{F}$ as a subgraph. Let $\mathcal{K}_{\ell + 1}^{r}$ be the collection of all $r$-graphs $F$ with at most $\binom{\ell+1}{2}$ edges such that for some $\left(\ell+1\right)$-set $S$ every pair $\{u, v\} \subset S$ is covered by an edge in $F$. Our first result shows that for each $\ell \geq r \geq 2$ every $\mathcal{K}_{\ell+1}^{r}$-saturated $r$-graph on $n$ vertices with $t_{r}(n, \ell) - o(n^{r-1+1/\ell})$ edges contains a complete $\ell$-partite subgraph on $(1-o(1))n$ vertices, which extends a…
Asymmetric induced saturation
研究对于哪些图H存在至少有一条边且无H诱导子图的图G，使得删除G的任意边都会产生H的诱导副本，并证明该存在性对多种图类成立。
A graph $G$ is called $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but removing any edge from $G$ creates an induced copy of $H$ and adding any edge of $G^{c}$ to $G$ creates an induced copy of $H$. Martin and Smith showed that there does not exist a $P_{4}$-induced-saturated graph, where $P_{4}$ is the path on 4 vertices. Axenovich and Csikós studied related questions, and asked if there exists a $P_{n}$-induced-saturated graph for any $n\geq5$. Our aim in this short note is to show that there exists a $P_{6}$-induced-saturated graph.
A graph G is uniquely K_r-saturated if it contains no clique with r vertices and if for all edges e in the complement, G + e has a unique clique with r vertices. Previously, few examples of uniquely K_r-saturated graphs were known, and little was known about their properties. We search for these graphs by adapting orbital branching, a technique originally developed for symmetric integer linear programs. We find several new uniquely K_r-saturated graphs with 4 \leq r \leq 7, as well as two new infinite families based on Cayley graphs for Z_n with a small number of generators.
Saturated Partial Embeddings of Planar Graphs
本文定义平面图的平面饱和率，证明无孪生平面图的该比率严格大于1/16且可任意逼近1/16，并研究含受限低度孪生顶点的更广泛图类。
Given a graph $H$, we say that a graph $G$ is $H$-saturated if $G$ contains no copy of $H$ but adding any new edge to $G$ creates a copy of $H$. Let $sat(n,K_r,t)$ be the minimum number of edges in a $K_r$-saturated graph on $n$ vertices with minimum degree at least $t$. Day showed that for fixed $r \geq 3$ and $t \geq r-2$, $sat(n,K_r,t)=tn-c(r,t)$ for large enough $n$, where $c(r,t)$ is a constant depending on $r$ and $t$, and proved the bounds $$ 2^t t^{3/2} \ll_r c(r,t) \leq t^{t^{2t^2}} $$ for fixed $r$ and large $t$. In this paper we show that for fixed $r$ and large $t$, the order of ma…
A simple topological graph is a topological graph in which any two edges have at most one common point, which is either their common endpoint or a proper crossing. More generally, in a k-simple topological graph, every pair of edges has at most k common points of this kind. We construct saturated simple and 2-simple graphs with few edges. These are k-simple graphs in which no further edge can be added. We improve the previous upper bounds of Kynčl, Pach, Radoičić, and Tóth and show that there are saturated simple graphs on n vertices with only 7n edges and saturated 2-simple graphs on n vertic…
There are no nontrivial uniquely C_9-saturated graphs
本文证明了不存在非平凡的唯C_9-饱和图，通过按最长偶圈长度分类并辅以计算机枚举验证，解决了Wenger-West猜想在t=9的情形。
We say $G$ is \emph{$(Q_n,Q_m)$-saturated} if it is a maximal $Q_m$-free subgraph of the $n$-dimensional hypercube $Q_n$. A graph, $G$, is said to be $(Q_n,Q_m)$-semi-saturated if it is a subgraph of $Q_n$ and adding any edge forms a new copy of $Q_m$. The minimum number of edges a $(Q_n,Q_m)$-saturated graph (resp. $(Q_n,Q_m)$-semi-saturated graph) can have is denoted by $sat(Q_n,Q_m)$ (resp. $s\text{-}sat(Q_n,Q_m)$). We prove that $ \lim_{n\to\infty}\frac{sat(Q_n,Q_m)}{e(Q_n)}=0$, for fixed $m$, disproving a conjecture of Santolupo that, when $m=2$, this limit is $\frac{1}{4}$. Further, we s…
Impact of local girth on the S-packing coloring of k-saturated subcubic graphs
研究局部圈长对k饱和子三次图S-打包着色能力的影响，通过参数分析和构造验证，揭示饱和度与局部圈长共同决定的可接受S-打包序列。
On the number of edges in saturated partial embeddings of maximal planar graphs
本文研究极大平面图饱和部分平面嵌入的边数极值问题，证明平面饱和数存在严格小于3倍顶点数的上界，并确定其与边数比值的最小范围，回答了Clifton和Simon提出的问题。
Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number and extremal number of $\mathcal{F}$, denoted $sat(n,\mathcal{F})$ and $ex(n,\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\mathcal{F}$-saturated graphs. For $k\in\mathbb{N}$, let $\mathcal{F}_k$ and $\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n…
Twin-free $K_r$-saturated Graphs and Maximally Independent Sets in $K_3$-free Graphs
针对无孪生顶点的$K_r$-饱和图问题，定义$tsat(n,K_r)$为其最小边数并给出上下界，同时揭示其与无$K_3$图中极大独立集相关问题的联系。
Weakly saturated random graphs
针对弱H饱和图的定义，本研究给出所有图H对应的Erdős-Rényi图𝒢ₙ,p弱H饱和临界阈值p_c的渐近下界，以及严格平衡H的上界，改进了H=Kᵣ的上界并推测其紧性。
For a graph $F$, we say that another graph $G$ is $F$-saturated, if $G$ is $F$-free and adding any edge to $G$ would create a copy of $F$. We study for a given graph $F$ and integer $n$ whether there exists a regular $n$-vertex $F$-saturated graph, and if it does, what is the smallest number of edges of such a graph. We mainly focus on the case when $F$ is a complete graph and prove for example that there exists a $K_3$-saturated regular graph on $n$ vertices for every large enough $n$. We also study two relaxed versions of the problem: when we only require that no regular $F$-free supergraph …
Given graphs $G$ and $H$, $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but for all $e \notin E(G)$, $H$ appears as a subgraph of $G + e$. While for every $n \ge |V(H)|$, there exists an $n$-vertex graph that is $H$-saturated, the same does not hold for induced subgraphs. That is, there exist graphs $H$ and values of $n \ge |V(H)|$ for which every $n$-vertex graph $G$ either contains $H$ as an induced subgraph, or there exists $e \notin E(G)$ such that $G + e$ does not contain $H$ as an induced subgraph. To circumvent this, Martin and Smith make use of trigraphs when introducing the co…
Saturated Partial Embeddings of Maximal Planar Graphs
本文研究极大平面图部分嵌入的标号与无标号两种饱和概念，给出标号平面饱和比的近紧上下界，并构造无标号平面饱和比至少为 $1/2$ 的图族。
Many interesting graph families contain only 2-connected graphs, which have ear decompositions. We develop a technique to generate families of unlabeled 2-connected graphs using ear augmentations and apply this technique to two problems. In the first application, we search for uniquely K_r-saturated graphs and find the list of uniquely K_4-saturated graphs on at most 12 vertices, supporting current conjectures for this problem. In the second application, we verifying the Edge Reconstruction Conjecture for all 2-connected graphs on at most 12 vertices. This technique can be easily extended to m…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。