Max Aires
The saturation number $\text{sat}(n,\mathcal{F})$ is the minimum number of edges in any graph which does not contain a member of $\mathcal{F}$ as a subgraph, but will if any edge is added. We give a few upper and lower bounds for saturation numbers for minors. In particular, we shall show that certain Generalized Petersen Graphs are $K^r$-minor saturated for $6\le r\le 8$.
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Degreewise Cut-Semigroup Saturation for $K_5$-Minor-Free Graphs and Seymour's Planar Edge-Colouring Conjecture
该论文证明了 $K_5$-无 minors 图的齐次割半群在高度 $k$ 的饱和性与平面 $k$-图的 $k$-边可染色性等价，从而将割多面体正规性猜想与 Seymour 平面边染色猜想联系起来。
The saturation number of a graph is the cardinality of a smallest maximal matching. This paper presents bounds for the saturation number of carbon nanocones which are asymptotically equal. The same techniques are applied for the saturation number of certain families of carbon nanotubes, which improve previous results and in one case, yields the exact value.
The "minor crossing number" of a graph $G$ is the minimum crossing number of a graph that contains $G$ as a minor. It is proved that for every graph $H$ there is a constant $c$, such that every graph $G$ with no $H$-minor has minor crossing number at most $c|V(G)|$.
For a given graph $F$, the $F$-saturation number of a graph $G$ is the minimum number of edges in an edge-maximal $F$-free subgraph of $G$. Recently, the $F$-saturation number of the Erdős$\text{\bf--}$Rényi random graph $\mathbbmsl{G}(n, p)$ has been determined asymptotically for any complete graph $F$. In this paper, we give an asymptotic formula for the $F$-saturation number of $\mathbbmsl{G}(n, p)$ when $F$ is a star graph.
We introduce a notion of rainbow saturation and the corresponding rainbow saturation number. This is the saturation version of the rainbow Turán numbers whose systematic study was initiated by Keevash, Mubayi, Sudakov, and Verstraëte. We give examples of graphs for which the rainbow saturation number is bounded away from the ordinary saturation number. This includes all complete graphs $K_n$ for $n\geq 4$, and several bipartite graphs. It is notable that there are non-bipartite graphs for which this is the case, as this does not happen when it comes to the rainbow extremal number versus the tr…
The saturation number of a graph $G$ is the cardinality of any smallest maximal matching of $G$, and it is denoted by $s(G)$. Fullerene graphs are cubic planar graphs with exactly twelve 5-faces; all the other faces are hexagons. They are used to capture the structure of carbon molecules. Here we show that the saturation number of fullerenes on $n$ vertices is essentially $n/3$.
Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The cardinality of any smallest maximal matching in $G$ is the saturation number of $G$ and is denoted by $s(G)$. In this paper we study the saturation number of the corona product of two specific graphs. We also consider some graphs with certain constructions that are of importance in chemistry and study their saturation number.
A graph $G$ is $H$-saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$. The saturation number, $sat (n, H)$, is the minimum number of edges of a graph in the set of all $H$-saturated graphs with order $n$. In this paper, we determine the saturation number $sat (n, P_6 + tP_2)$ for $n \geq 10t/3 + 10$ and characterize the extremal graphs for $n >10t/3 + 20$.
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, …
A Note on Weak Saturation Number of Trees
本文研究树的弱饱和数，以毛毛虫为例，给出了若干紧致估计，并证明对于任意α∈[1,2]，存在k个顶点的毛毛虫其弱饱和数为k^α量级。还提出好树的定义并给出充分条件。
Given a poset $P$, a family $F$ of elements in the Boolean lattice is said to be $P$-saturated if (1) $F$ contains no copy of $P$ as a subposet and (2) every proper superset of $F$ contains a copy of $P$ as a subposet. The maximum size of a $P$-saturated family is denoted by $La(n,P)$, which has been studied for a number of choices of $P$. The minimum size of a $P$-saturated family, $sat(n,P)$, was introduced by Gerbner et al. (2013), and parallels the deep literature on the saturation function for graphs. We introduce and study the concept of saturation for induced subposets. As opposed to in…
The Hadwiger number $h(G)$ is the order of the largest complete minor in $G$. Does sufficient Hadwiger number imply a minor with additional properties? In [2], Geelen et al showed $h(G)\geq (1+o(1))ct\sqrt{\ln t}$ implies $G$ has a bipartite subgraph with Hadwiger number at least $t$, for some explicit $c\sim 1.276\dotsc$. We improve this to $h(G) \geq (1+o(1))t\sqrt{\log_2 t}$, and provide a construction showing this is tight. We also derive improved bounds for the topological minor variant of this problem.
Motivated by Hadwiger's conjecture, we prove that every $n$-vertex graph $G$ with no independent set of size three contains an $\lceil n/2\rceil$-vertex simple minor $H$ with $$0.98688 \cdot \binom{|V(H)|}{2} - o(n^2)$$ edges.
We derive precise asymptotic estimates for the number of labelled graphs not containing $K_{3,3}$ as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random $K_{3,3}$-minor-free graphs, like the expected number of edges. To establish these results, we translate a decomposition for the corresponding graph class into equations for generating functions and use singularity analysis. We also find a precise estimate for the number of graphs not containing the graph $K_{3,3}$ plus an edge as a minor.
In this paper, we discuss a generalization of the notion of saturation in graphs in order to deal with induced structures. In particular, we define ${\rm indsat}(n,H)$, which is the fewest number of gray edges in a trigraph so that no realization of that trigraph has an induced copy of $H$, but changing any white or black edge to gray results in some realization that does have an induced copy of $H$. We give some general and basic results and then prove that ${\rm indsat}(n,P_4)=\lceil (n+1)/3\rceil$ for $n\geq 4$ where $P_4$ is the path on 4 vertices. We also show how induced saturation in th…
The rainbow saturation number is linear
该研究证明任意非空图的彩虹饱和数关于顶点数n是线性的，证实了Girão等人的一个猜想，同时给出完全图彩虹饱和数的改进上界，证否了该团队的另一猜想。
Treewidth versus clique number: induced minors
本文证明遗传图类的树宽-团数有界性等价于其诱导子式类的树宽-团数有界性。
Some results on the saturation number for unions of cliques
该研究属于组合数学领域，确定了三类团并图的饱和数及部分极图，完善了图饱和问题的相关结论。
Let ${\rm ex \,} {\mathcal B}$ be a minor-closed class of graphs with a set ${\mathcal B}$ of minimal excluded minors. We study (a) the asymptotic number of graphs without $k+1$ disjoint minors in ${\mathcal B}$ and (b) the properties of a uniformly random graph drawn from all such graphs on vertices $\{1,\dots,n\}$. We present new results in the case when ${\rm ex \,} {\mathcal B}$ contains arbitrarily large fans for a general (good enough) set of forbidden minors ${\mathcal B}$. A particular case where our results hold is ${\mathcal B} = \{K_4\}$. For any fixed $k = 1, 2, \dots$ we derive pr…
Hadwiger's conjecture implies that $n\leαh$ for all graphs of order $n$, stability number $α$, and Hadwiger number $h$. Combining ideas of Kawarabayashi et al. and Wood, we prove that $n\le (α-1)(2h-5)+5$ for such graphs if $α\ge 3$ and $h\ge 5$.
A graph $G$ is called $F$-saturated if $G$ does not contain $F$ as a subgraph (not necessarily induced) but the addition of any missing edge to $G$ creates a copy of $F$. The saturation number of $F$, denoted by $sat(n,F)$, is the minimum number of edges in an $n$-vertex $F$-saturated graph. Determining the saturation number of complete partite graphs is one of the most important problems in the study of saturation number. The value of $sat(n,K_{2,2})$ was shown to be $\lfloor\frac{3n-5}{2}\rfloor$ by Ollmann, and a shorter proof was later given by Tuza. For $K_{2,3}$, there has been a series …
We show that, if $k$ and $\ell$ are positive integers and $r$ is sufficiently large, then the number of rank-$k$ flats in a rank-$r$ matroid $M$ with no $U_{2,\ell+2}$-minor is less than or equal to number of rank-$k$ flats in a rank-$r$ projective geometry over GF$(q)$, where $q$ is the largest prime power not exceeding $\ell$.
In this paper, we show the following three theorems. Let $G$ be a $K_5$-minor-free graph. Then Alon-Tarsi number of $G$ is at most $5$, there exists a matching $M$ of $G$ such that the Alon-Tarsi number of $G-M$ is at most $4$, and there exists a forest $F$ such that the Alon-Tarsi number of $G-E(F)$ is at most $3$.
Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log t})$-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no $K_t$ minor, we make the first improvement on this bound by showing that every graph with no odd $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$.
We show that, if $q$ is a prime power at most 5, then every rank-$r$ matroid with no $U_{2,q+2}$-minor has no more lines than a rank-$r$ projective geometry over GF$(q)$. We also give examples showing that for every other prime power this bound does not hold.
Saturation for Sums of Posets and Antichains
本文研究了偏序集和反链的线性和的饱和数，证明其至少为线性，扩展了钻石偏序集的结论。
Given a graph $F$, a hypergraph is a Berge-$F$ if it can be obtained by expanding each edge in $F$ to a hyperedge containing it. A hypergraph $H$ is Berge-$F$-saturated if $H$ does not contain a subgraph that is a Berge-$F$, but for any edge $e\in E(\overline{H})$, $H+e$ does. The $k$-uniform saturation number of Berge-$F$ is the minimum number of edges in a $k$-uniform Berge-$F$-saturated hypergraph on $n$ vertices. For $k=2$ this definition coincides with the classical definition of saturation for graphs. In this paper we study the saturation numbers for Berge triangles, paths, cycles, stars…
A convex geometric hypergraph (abbreviated cgh) consists of a collection of subsets of a strictly convex set of points in the plane. Extremal problems for cgh's have been extensively studied in the literature, and in this paper we consider their corresponding saturation problems. We asymptotically determine the saturation number of two geometrically disjoint $r$-tuples. Further, amongst the eight nonisomorphic $3$-uniform cgh's on two edges, we determine the saturation number for seven of these up to order of magnitude and the eighth up to a log factor.
Proper rainbow saturation for trees
本文系统研究树的恰当彩虹饱和数，针对若干无限族树给出了精确与渐近结果，并揭示了其与经典饱和数及半饱和数的联系。
There are only a finite number of excluded minors for the class of bicircular matroids
本文证明了双圈拟阵类仅有有限个排除子式，核心方法是利用偏置图和拟图拟阵表示，证明秩至少为十的排除子式必为拟图拟阵并包含已知小排除子式，从而得出结论。
On saturation numbers of complete multipartite graphs and even cycles
本文研究完全多部图与足够长偶圈对应的图饱和数的渐近行为，拓展了已有结果并部分解决了相关猜想。
Odd complete bipartite minors in graphs with independence number two
本文将 Chen 和 Deng 关于独立数为二的图包含完全二部图子式的结果推广到奇子式情形，证明了独立数为二的图必包含相应的奇完全二部图子式。
A note on the saturation number for unions of three cliques
本文针对2≤p≤q<r<p+q且n足够大的情况，确定了三个团K_p、K_q、K_r的并的饱和数及唯一极图，结合已有结果完成了所有足够大n下该类并的饱和数及极图的确定。
The dichromatic number $\vecχ(D)$ of a digraph $D$ is the smallest $k$ for which it admits a $k$-coloring where every color class induces an acyclic subgraph. Inspired by Hadwiger's conjecture for undirected graphs, several groups of authors have recently studied the containment of directed graph minors in digraphs with given dichromatic number. In this short note we improve several of the existing bounds and prove almost linear bounds by reducing the problem to a recent result of Postle on Hadwiger's conjecture.
Given graphs $H$ and $F$, a subgraph $G\subseteq H$ is an $F$-saturated subgraph of $H$ if $F\nsubseteq G$, but $F\subseteq G+e$ for all $e\in E(H)\setminus E(G)$. The saturation number of $F$ in $H$, denoted $\text{sat}(H,F)$, is the minimum number of edges in an $F$-saturated subgraph of $H$. In this paper we study saturation numbers of tripartite graphs in tripartite graphs. For $\ell\ge 1$ and $n_1$, $n_2$, and $n_3$ sufficiently large, we determine $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell})$ and $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-1})$ exactly and $\text{sat}(K_{n_1,n_2,n_3…
The Saturation Number for the Diamond is Linear
本文证明了钻石偏序集的饱和数下界为(n+1)/5，证实其饱和数是线性的。
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…
Interval minors of bipartite graphs were introduced by Jacob Fox in the study of Stanley-Wilf limits. Recently, Mohar, Rafiey, Tayfeh-Rezaie and Wu investigated the maximum number of edges in $K_{k,\ell}$-interval minor free bipartite graphs when $k=2$ and $k=3$. In this paper, we investigate the maximum number of edges in $K_{k,\ell}$-interval minor free bipartite graphs for general $k$ and $\ell$. We also study the maximum number of edges in $K_{\ell_1,\ell_2,\cdots,\ell_t}$-interval minor free multipartite graphs.
The graph minor structure theorem by Robertson and Seymour shows that every graph that excludes a fixed minor can be constructed by a combination of four ingredients: graphs embedded in a surface of bounded genus, a bounded number of vortices of bounded width, a bounded number of apex vertices, and the clique-sum operation. This paper studies the converse question: What is the maximum order of a complete graph minor in a graph constructed using these four ingredients? Our main result answers this question up to a constant factor.
Saturation Numbers for Linear Forests $P_7+tP_2$
该研究针对图论中的饱和数问题，确定了线性森林$P_7+tP_2$在$n$满足一定下界时的饱和数取值，并刻画了对应条件下的极图结构。
For any positive integer $l$ we prove that if $M$ is a simple matroid with no $(l+2)$-point line as a minor and with sufficiently large rank, then $|E(M)|\le \frac{q^{r(M)}-1}{q-1}$, where $q$ is the largest prime power less than or equal to $l$. Equality is attained by projective geometries over GF$(q)$.
The Alon-Tarsi number of $K_{3,3}$-minor-free graphs
本文证明$K_{3,3}$-无minor图的Alon-Tarsi数至多为5，且存在匹配和森林使其分别降至4和3，推广了相关结果。
We prove that if $G$ is a graph with an minimal edge cut $F$ of size three and $G_1$, $G_2$ are the two (augmented) components of $G-F$, then the crossing number of $G$ is equal to the sum of crossing numbers of $G_1$ and $G_2$. Combining with known results, this implies that crossing number is additive over edge-cuts of size $d$ for $d\in\{0, 1, 2, 3\}$, whereas there are counterexamples for every $d\ge 4$. The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces…
Hadwiger's conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colorable. It is known that the exact version of Hadwiger's conjecture does not extend to list coloring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant $c$ such that every graph with no $K_t$-minor has list chromatic number at most $ct$. More specifically, they also conjectured that this holds for $c=\frac{3}{2}$. Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no $K_t$-minor is at least $(2-o(1))t$, and hence $c \ge 2$ in the…
Odd minors or odd immersions in graphs with independence number two
针对独立数为2的图，在Kühn等人反驳奇Hadwiger猜想的背景下，证明这类图必包含K_{⌈n/2⌉}作为奇子式或全奇浸入，完善了相关图论结论。
On the proper rainbow saturation numbers of cliques, paths, and odd cycles
本文研究了图的正常彩虹饱和数，确定了路及 $K_4$ 的该数值，并对更大团、特定树和奇圈的情况给出了界。
Let $F$ and $H$ be $k$-uniform hypergraphs. We say $H$ is $F$-saturated if $H$ does not contain a subgraph isomorphic to $F$, but $H+e$ does for any hyperedge $e\not\in E(H)$. The saturation number of $F$, denoted $\mathrm{sat}_k(n,F)$, is the minimum number of edges in a $F$-saturated $k$-uniform hypergraph $H$ on $n$ vertices. Let $C_3^{(3)}$ denote the $3$-uniform loose cycle on $3$ edges. In this work, we prove that \[ \left(\frac{4}3+o(1)\right)n\leq \mathrm{sat}_3(n,C_3^{(3)})\leq \frac{3}2n+O(1). \] This is the first non-trivial result on the saturation number for a fixed short hypergra…
Graph $G$ is $F$-saturated if $G$ contains no copy of graph $F$ but any edge added to $G$ produces at least one copy of $F$. One common variant of saturation is to remove the former restriction: $G$ is $F$-semi-saturated if any edge added to $G$ produces at least one new copy of $F$. In this paper we take this idea one step further. Rather than just allowing edges of $G$ to be in a copy of $F$, we require it: $G$ is $F$-covered if every edge of $G$ is in a copy of $F$. It turns out that there is smooth interaction between coverage and semi-saturation, which opens for investigation a natural an…
We compute the number of all rhombus tilings of a hexagon with sides $a,b+1,c,a+1,b,c+1$, of which the central triangle is removed, provided $a,b,c$ have the same parity. The result is a product of four numbers, each of which counts the number of plane partitions inside a given box. The proof uses nonintersecting lattice paths and a new identity for Schur functions, which is proved by means of the minor summation formula of Ishikawa and Wakayama. A symmetric generalization of this identity is stated as a conjecture.
On the saturation number of the kite graph
本文研究风筝图的饱和数，建立其与连通性的关系，确定该饱和数并刻画极值图，部分回答了Hua和Peng的相关问题。
The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We prove that for every graph $H$, the clustered chromatic number of the class of $H$-minor-free graphs is tied to the tree-depth of $H$. In particular, if $H$ is connected with tree-depth $t$ then every $H$-minor-free graph is $(2^{t+1}-4)$-colourable with monochromatic components of size at most $c(H)$. This provides the first evidence for a conjecture of Ossona de Mendez, Oum and Wood (20…
The saturation number of $C_6$
本文针对极值图论中$C_6$饱和数的公开问题，推导得出$n\ge9$时$sat(n,C_6)$的新上下界，大幅提升了该参数的已知界精度。
Gluing Posets and the Dichotomy of Poset Saturation Numbers
本文通过定义偏序集的“粘合”操作，证明该操作保持饱和数的有界与无界性，并应用于证明任何偏序集最多添加3个元素后其饱和数增长至多为线性。
Evolution of random representable matroids: minors, circuits, connectivity and the critical number
本文研究了由有限域上随机矩阵序列表示的随机拟阵演化过程，分析了幼阵、圈、连通性与临界数的出现规律，并解决了文献中的若干开放问题。
The saturation number of wheels
本文研究了轮图$W_k$的（半）饱和数，证明了在$k \geq 8$且$n \geq 56k^3$时，$(s)sat(n,W_k)=n-1+(s)sat(n-1,C_k)$，并给出了最大度限制下的半饱和数下界。
A fan $F_k$ is a graph that consists of an induced path on $k$ vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers $q$ and $k$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a vertex-minor isomorphic to $F_k$. We also prove that for all positive integers $q$ and $k\ge 3$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a pivot-minor isomorphic to a cycle of length $k$.
We prove that graphs excluding a fixed immersion have bounded nonrepetitive chromatic number. More generally, we prove that if $H$ is a fixed planar graph that has a planar embedding with all the vertices with degree at least 4 on a single face, then graphs excluding $H$ as a topological minor have bounded nonrepetitive chromatic number. This is the largest class of graphs known to have bounded nonrepetitive chromatic number.
Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $G$ is called $\mathcal{F}$-saturated if it does not contain any members of $\mathcal{F}$ but adding any edge creates a copy of some $r$-graph in $\mathcal{F}$. The saturation number $\operatorname{sat}(\mathcal{F},n)$ is the minimum number of edges in an $\mathcal{F}$-saturated graph on $n$ vertices. We prove that there exists a finite family $\mathcal{F}$ such that $\operatorname{sat}(\mathcal{F},n) / n^{r-1}$ does not tend to a limit. This settles a question of Pikhurko.
Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has l…
The $t$-colored rainbow saturation number $rsat_t(n,F)$ is the minimum size of a $t$-edge-colored graph on $n$ vertices that contains no rainbow copy of $F$, but the addition of any missing edge in any color creates such a rainbow copy. Barrus, Ferrara, Vandenbussche and Wenger conjectured that $rsat_t(n,K_s) = Θ(n\log n)$ for every $s\ge 3$ and $t\ge \binom{s}{2}$. In this short note we prove the conjecture in a strong sense, asymptotically determining the rainbow saturation number for triangles. Our lower bound is probabilistic in spirit, the upper bound is based on the Shannon capacity of a…
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