The canonical tree-decomposition theorem, proved by Robertson and Seymour in their seminal graph minors series, turns out to be an extremely valuable tool in structural and algorithmic graph theory. In this paper, we prove the analogous result for digraphs, the directed tangle tree-decomposition theorem. More precisely, we introduce directed tangles and provide a directed tree-decomposition of digraphs $G$ that distinguishes all maximal directed tangles in $G$. Furthermore, for any integer $k$, we construct a directed tree-decomposition that distinguishes all directed tangles of order $k$, for any integer $k$.

By relaxing the bound slightly, we can make the previous result algorithmic: for fixed $k$, we design a polynomial-time algorithm that finds a directed tree-decomposition distinguishing all directed tangles of order $3k$ separated by some separation of order less than $k$.

We provide two direct applications of this tangle tree-decomposition theorem. First, we show that the family of directed odd cycles has the half-integral Erdős-Pósa property, that is, there is a function $f:\mathbb{N}\rightarrow \mathbb{R}$ such that for every digraph $G$ and every integer $k$, either $G$ contains a family of $k$ directed odd cycles where every vertex of $G$ is contained at most two cycles, or a vertex subset of size at most $f(k)$ hitting all directed odd cycles. This extends the half-integral Erdős-Pósa property for undirected odd cycles, shown by Reed [Mangoes and blueberries. Combinatorica 1999].

Second, for every fixed $k$ we show that there is a polynomial-time algorithm which, on input $G$, and source and sink vertices $(s_1, t_1), \dots, (s_k, t_k)$, either finds a family of paths $P_1, \dots, P_k$ such that each $P_i$ links $s_i$ to $t_i$ and every vertex of $G$ is contained in at most two paths, or determines that there is no set of pairwise vertex-disjoint paths each connecting $s_i$ to $t_i$. This result improves previous results (with “two” replaced by “three”), and given known hardness results, our result is best possible in a sense that we cannot hope for fixed parameter tractability or fully vertex-disjoint directed paths.

This is joint work with Archontia C. Giannopoulou, Ken-ichi Kawarabayashi, Stephan Kreutzer, and Qiqin Xie.

On May 19, 2020, O-joung Kwon (권오정) from Incheon National University and IBS Discrete Mathematics Group presented a survey talk on the MIM-width of graphs. The title of his talk was “Mim-width: a width parameter beyond rank-width“.

Note added on June 2020 After I presented the talk, Benjamin Bergougnoux commented me that the claim of (Boyaci, Ekim, Shalom 17) at 1:02:40 was recently disproved by Kratochvíl, Masařík, and Novotná (https://arxiv.org/abs/2002.08311). I would like to thank Benjamin for the comment. Whether or not there is a polynomial-time algorithm for Max Cut on proper interval graphs is still open.

Vatshelle (2012) introduced a width parameter called mim-width. It is based on the following cut function : for a vertex partition (A,B) of a graph, the complexity of this partition is computed by the size of a maximum induced matching of the bipartite subgraph induced by edges between A and B. This parameter naturally extends the expressibility power of the graph parameters clique-width and rank-width, which have been well-developed in recent years. In a series of papers, we explored the computational complexity of several problems, parameterized by mim-width. We summarize known structural properties and algorithmic applications of mim-width, and give some open problems at the end. This is joint work with Lars Jaffke, Torstein Strømme, and Jan Arne Telle.

The 2nd East Asia Workshop on Extremal and Structural Graph Theory is a workshop to bring active researchers in the field of extremal and structural graph theory, especially in the East Asia such as China, Japan, and Korea.

Date

Oct 31, 2019 (Arrival Day) – Nov 4, 2019 (Departure Day)

Venue and Date

1st floor Diamond Hall

UTOP UBLESS Hotel, Jeju, Korea (유탑유블레스호텔제주) Address: 502 Johamhaean-ro, Jocheon-eup, Jeju, Korea (제주특별자치도 제주시 조천읍 조함해안로 502) We plan to support the accommodation for invited participants.

Invited Speakers

Ping Hu, Sun Yat-Sen University, China

Jaehoon Kim, KAIST, Korea

O-joung Kwon, Incheon National University and IBS Discrete Mathematics Group, Korea

Joonkyung Lee, University of Hamburg, Germany

Binlong Li, Northwestern Polytechnical University, China

Hongliang Lu, Xi’an Jiaotong University, China

Abhishek Methuku, IBS Discrete Mathematics Group, Korea

Atsuhiro Nakamoto, Yokohama National University, Japan

Kenta Noguchi, Tokyo University of Science, Japan

Kenta Ozeki, Yokohama National University, Japan

Boram Park, Ajou University, Korea

Yuejian Peng, Hunan University, China

Zi-Xia Song, University of Central Florida, U.S.A.

Tomáš Kaiser, University of West Bohemia, Czech Republic.

Maho Yokota, Tokyo University of Science, Japan.

Xuding Zhu, Zhejiang Normal University, China

More speakers to be announced as soon as confirmed. Last update: September 10.

Program

Day 0 (Oct. 31 Thursday)

4:00PM-6:00Pm Registration and Discussions

Day 1 (Nov. 1 Friday)

9:00AM-9:20AM Opening address

9:20AM-9:50AM Jaehoon Kim, A quantitative result on the polynomial Schur’s theorem

10:00AM-10:30AM Yuejian Peng, Lagrangian densities of hypergraphs

10:30AM-10:50AM Coffee Break

10:50AM-11:20AM Atsuhiro Nakamoto, Geometric quadrangulations on the plane

11:30AM-12:00PM Ping Hu, The inducibility of oriented stars

2:00PM-2:30PM Boram Park, 5-star coloring of some sparse graphs

2:40PM-3:10PM Kenta Ozeki, An orientation of graphs with out-degree constraint

3:10PM-3:30PM Coffee Break

3:30PM-5:30PM Problem session

Day 2 (Nov. 2 Saturday)

9:20AM-9:50AM Xuding Zhu, List colouring and Alon-Tarsi number of planar graphs

10:00AM-10:30AM O-joung Kwon, A survey of recent progress on Erdős-Pósa type problems

10:30AM-10:50AM Coffee Break

10:50AM-11:20AM Kenta Noguchi, Extension of a quadrangulation to triangulations, and spanning quadrangulations of a triangulation

11:30AM-12:00PM Zi-Xia Song, Ramsey numbers of cycles under Gallai colorings

2:00PM-2:30PM Binlong Li, Cycles through all finite vertex sets in infinite graphs

2:40PM-3:10PM Tomáš Kaiser, Hamilton cycles in tough chordal graphs

3:20PM-3:50PM Abhishek Methuku, On a hypergraph bipartite Turán problem

3:50PM-4:10PM Coffee Break

4:10PM-6:00PM Problem session and discussion

Day 3 (Nov. 3 Sunday)

9:20AM-9:50AM Joonkyung Lee, Odd cycles in subgraphs of sparse pseudorandom graphs

10:00AM-10:30AM Maho Yokota, Connectivity, toughness and forbidden subgraph conditions

10:30AM-10:50AM Coffee Break

10:50AM-11:20AM Hongliang Lu, On minimum degree thresholds for fractional perfect matchings and near perfect matchings in hypergraphs

11:30AM-12:00PM Contributed Talks

2:00PM-6:00PM Problem session / Discussions / Hike

Held at and sponsored by Shanghai Center for Mathematical Sciences in China, under the name “2018 SCMS Workshop on Extremal and Structural Graph Theory”.

Organizers: Ping Hu, Seog-Jin Kim, Kenta Ozeki, Hehui Wu.

Let $S_{k,\ell}$ denote the oriented star with $k+\ell$ edges, where the center has out-degree $k$ and in-degree $\ell$. For all $k,\ell$ with $k+\ell$ large, we determine n-vertex digraphs $G$ which maximize the number of induced $S_{k,\ell}$. This extends a result of Huang (2014) for all $S_{k,0}$, and a result of Hladký, Král’ and Norin for $S_{1,1}$. Joint work with Jie Ma, Sergey Norin, and Hehui Wu.

Jaehoon Kim, A quantitative result on the polynomial Schur’s theorem

Recently, Liu, Pach, and Sándor [arXiv:1811.05200] proved that for a polynomial $p(z)\in \mathbb{Z}[z]$, any $2$-coloring of $\mathbb{N}$ has infinitely many monochromatic solutions of the equatoin $x+y=p(z)$ if and only if $2\mid p(0)p(1)$. We improve their result in a quantitative way. We prove that if $p(z)$ has degree $d \neq 3$ and $2\mid p(0)p(1)$, then any $2$-coloring of $[n]=\{1,\dots, n\}$ contains at least $n^{2/d^2 -o(1)}$ monochromatic solutions. This is sharp as there exists a coloring of $[n]$ with $O(n^{2/d^2})$ monochromatic solutions. Our method also gives some bound for the case when $d=3$, but it is not sharp. We also prove that if $2\mid p(0)p(1)$, then the interval $[n, p(\lceil \frac{p(n)}{2} \rceil)]$ contains at least one monochromatic solution of $x+y=p(z)$. This is sharp up to multiplicative constant at most two as one can color $[n, \frac{1}{2}p(\lceil \frac{p(n)}{2} \rceil)-1]$ with no monochromatic solutions. Joint work with Hong Liu and Péter Pál Pach.

O-joung Kwon, A survey of recent progress on Erdős-Pósa type problems

A graph family $\mathcal{F}$ is said to have the Erdős-Pósa property if there is a function $f$ such that for every graph $G$ and an integer $k$, either $G$ contains $k$ disjoint copies of graphs in $\mathcal{F}$, or it has a vertex set of size at most $f(k)$ that hits all copies of graphs in $\mathcal{F}$. This name is motivated from the Erdős-Pósa theorem (1965) which says that the set of cycles has the Erdős-Pósa property. In this talk, we survey on progress of finding various graph families that have the Erdős-Pósa property, and would like to pose interesting open problems.

Joonkyung Lee, Odd cycles in subgraphs of sparse pseudorandom graphs

We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let $\Gamma$ be an $(n,d,\lambda)$-graph, i.e., $n$-vertex, $d$-regular graphs with all nontrivial eigenvalues in the interval $[-\lambda,\lambda]$. Krivelevich, Lee, and Sudakov conjectured that, whenever $\lambda^{2k-1}\ll d^{2k}/n$, every subgraph $G$ of $\Gamma$ with $(1/2+o(1))e(\Gamma)$ edges contains an odd cycle $C_{2k+1}$. Aigner-Horev, Hàn, and the third author proved a weaker statment by allowing an extra polylogarithmic factor in the assumption $\lambda^{2k-1}\ll d^{2k}/n$, but we completely remove it and hence settle the conjecture. This also generalises Sudakov, Szabo, and Vu’s Turán-type theorem for triangles. Secondly, we obtain a Ramsey multiplicity result for odd cycles. Namely, in the same range of parameters, we prove that every 2-edge-colouring of $\Gamma$ contains at least $(1-o(1))2^{-2k}d^{2k+1}$ monochromatic copies of $C_{2k+1}$. Both results are asymptotically best possible by Alon and Kahale’s construction of $C_{2k+1}$-free pseudorandom graphs. Joint work with Sören Berger, Mathias Schacht.

Binlong Li, Cycles through all finite vertex sets in infinite graphs

A closed curve in the Freudenthal compactification $|G|$ of an infinite locally finite graph $G$ is called a Hamiltoniancurve if it meets every vertex of $G$ exactly once (and hence it meets every end at least once). We prove that $|G|$ has a Hamiltonian curve if and only if every finite vertex set of $G$ is contained in a cycle of $G$. We apply this to extend a number of results and conjectures on finite graphs to Hamiltonian curves in infinite locally finite graphs. For example, Barnette’s conjecture (that every finite planar cubic 3-connected bipartite graph is Hamiltonian) is equivalent to the statement that every one-ended planar cubic 3-connected bipartite graph has a Hamiltonian curve. It is also equivalent to the statement that every planar cubic 3-connected bipartite graph with a nowhere-zero 3-flow (with no restriction on the number of ends) has a Hamiltonian curve. However, there are 7-ended planar cubic 3-connected bipartite graphs that do not have a Hamiltonian curve. Joint work with André Kündgen and Carsten Thomassen.

Hongliang Lu, On minimum degree thresholds for fractional perfect matchings and near perfect matchings in hypergraphs

We study degree conditions for the existence of large matchings and fractional perfect matching in uniform hypergraphs. Firstly, we give some sufficient conditions for $k$-graphs to have fractional perfect matching in terms of minimum degree. Secondly, we prove that for integers $k,l,n$ with $k\ge 3$, $k/2<l<k$, and $n$ large, if $H$ is a $k$-uniform hypergraph on $n$ vertices and $\delta_{l}(H)>{n-l\choose k-l}-{(n-l)-(\lceil n/k \rceil-2)\choose 2}$, then $H$ has a matching covering all but a constant number of vertices. When $l=k-2$ and $k\ge 5$, such a matching is near perfect and our bound on $\delta_l(H)$ is best possible. When $k=3$, with the help of an absorbing lemma of Hàn, Person, and Schacht, our proof also implies that $H$ has a perfect matching, a result proved by Kühn, Osthus, and Treglown and, independently, of Kahn. Joint work with Xingxing Yu and Xiaofan Yuan.

Abhishek Methuku, On a hypergraph bipartite Turán problem

Let $t$ be an integer such that $t\geq 2$. Let $K_{2,t}^{(3)}$ denote the triple system consisting of the $2t$ triples $\{a,x_i,y_i\}$, $\{b,x_i,y_i\}$ for $ 1 \le i \le t$, where the elements $a, b, x_1, x_2, \ldots, x_t,$ $y_1, y_2, \ldots, y_t$ are all distinct. Let $ex(n,K_{2,t}^{(3)})$ denote the maximum size of a triple system on $n$ elements that does not contain $K_{2,t}^{(3)}$. This function was studied by Mubayi and Verstraëte, where the special case $t=2$ was a problem of Erdős that was studied by various authors. Mubayi and Verstraëte proved that $ex(n,K_{2,t}^{(3)})<t^4\binom{n}{2}$ and that for infinitely many $n$, $ex(n,K_{2,t}^{(3)})\geq \frac{2t-1}{3} \binom{n}{2}$. These bounds together with a standard argument show that $g(t):=\lim_{n\to \infty} ex(n,K_{2,t}^{(3)})/\binom{n}{2}$ exists and that \[\frac{2t-1}{3}\leq g(t)\leq t^4.\] Addressing a 15 year old question of Mubayi and Verstraëte on the growth rate of $g(t)$, we prove that as $t \to \infty$, \[g(t) = \Theta(t^{1+o(1)}).\] Joint work with Beka Ergemlidze and Tao Jiang.

Atsuhiro Nakamoto, Geometric quadrangulations on the plane

Let $P$ be a point set on the plane with $|P| \geq 4$ in a general position (i.e., no three points lie on the same straight line). A geometric quadrangulation $Q$ on $P$ is a geometric plane graph (i.e., every edge is a straight segment) such that the outer cycle of $Q$ coincides with the boundary of the convex hull ${\rm Conv}(P)$ of $P$ and that each finite face of $Q$ is quadrilateral. We say that $P$ is quadrangulatable if $P$ admits a geometric quadrangulation. It is easy to see that if $P$ has an even number of points on the boundary of ${\rm Conv}(P)$, then $P$ is quadrangulatable. Suppose that $P$ is $k$-colored for $k \geq 2$, and that no two consecutive points on the boundary of ${\rm Conv}(P)$ have the same color. Let us consider whether $P$ is quadrangulatable with no edge joining two points with the same color. Then we see that $P$ is not necessarily quadrangulatable. Hence, introducing Steiner points $S$ for $P$, which are ones put in the interior of ${\rm Conv}(P)$ as we like, we consider whether $P \cup S$ is quadrangulatable. Intuitively, for any $k$-colored $P$, adding sufficiently large Steiner points $S$, we wonder if $P \cup S$ is quadrangulatable. However, we surprisingly see that it is impossible when $k=3$ (Alvarez et al., 2007). In my talk, we summarize these researches on quadrangulatability of point sets with Steiner points, and describe a relation with coloring of topological quadrangulations (Alvarez and Nakamoto, 2012 and Kato et al., 2014). Moreover, we describe a recent progress on a similar topic on quadrangulatability of a polygon with Steiner points.

Kenta Noguchi, Extension of a quadrangulation to triangulations, and spanning quadrangulations of a triangulation

A triangulation (resp., a quadrangulation) on a surface $S$ is a map of a graph (possibly with multiple edges and loops) on $S$ with each face bounded by a closed walk of length $3$ (resp., $4$). In this talk, we focus on the relationship between triangulations and quadrangulations on a surface. (I) An extension of a graph $G$ is the construction of a new graph by adding edges to some pairs of vertices in $G$. It is easy to see that every quadrangulation $G$ on any surface can be extended to a triangulation by adding a diagonal to each face of $G$. If we require the resulting triangulation to have more properties, the problem might be difficult and interesting. Our two main results are as follows. Every quadrangulation on any surface can be extended to an even (i.e. Eulerian) triangulation. Furthermore, we give the explicit formula for the number of distinct even triangulations extended from a given quadrangulation on a surface. These completely solves the problem raised by Zhang and He (2005). (II) It is easy to see that every loopless triangulation $G$ on any surface has a quadrangulation as a spanning subgraph of $G$. As well as (I), if we require the resulting quadrangulation to have more properties, the problem might be difficult and interesting. Kündgen and Thomassen (2017) proved that every loopless even triangulation $G$ on the torus has a spanning nonbipartite quadrangulation, and that if $G$ has sufficiently large face width, then $G$ also has a bipartite one. We prove that a loopless even triangulation $G$ on the torus has a spanning bipartite quadrangulation if and only if $G$ does not have $K_7$ as a subgraph. This talk is based on the papers (2015, 2019, 2019). Joint work with Atsuhiro Nakamoto and Kenta Ozeki.

Kenta Ozeki, An orientation of graphs with out-degree constraint

An orientation of an (undirected) graph $G$ is an assignment of directions to each edge of $G$. An orientation with certain properties has much attracted because of its applications, such as a list-coloring, and Tutte’s $3$-flow conjecture. In this talk, we consider an orientation such that the out-degree of each vertex is contained in a given list. For an orientation $O$ of $G$ and a vertex $v$, we denote by $d_O^+(v)$ the out-degree of $v$ in the digraph $G$ with respect to the orientation $O$. Recall that the number of outgoing edges is the out-degree. We denote by $\mathbb{N}$ the set of natural numbers (including $0$). For a graph $G$ and a mapping $L: V(G)\rightarrow 2^{\mathbb{N}}$, an orientation $O$ of $G$ such that \[d_O^+(v) \in L(v)\] for each vertex $v$ is called an $L$-orientation. In this talk, we pose the following conjecture. Conjecture. Let $G$ be a graph and let $L: V(G) \rightarrow 2^{\mathbb{N}}$ be a mapping. If \[|L(v)| \ \geq \ \frac{1}{2}\Big(d_G(v) +3\Big)\]for each vertex $v$, then $G$ has an $L$-orientation. I will explain some results related to Conjecture; the best possibility if it is true, and partial solutions for bipartite graphs. However, it is open even for complete graphs. This talk is based on the paper https://doi.org/10.1002/jgt.22498. Joint work with S. Akbari, M. Dalirrooyfard, K.Ehsani, and R. Sherkati.

Boram Park, 5-star coloring of some sparse graphs

A star $k$-coloring of a graph $G$ is a proper (vertex) $k$-coloring of $G$ such that the vertices on a path of length three receive at least three colors. Given a graph $G$, its star chromatic number, denoted $\chi_s(G)$, is the minimum integer $k$ for which $G$ admits a star $k$-coloring. Studying star coloring of sparse graphs is an active area of research, especially in terms of the maximum average degree $\mathrm{mad}(G)$ of a graph $G$. It is known that for a graph $G$, if $\mathrm{mad}(G)<\frac{8}{3}$, then $\chi_s(G)\leq 6$ (Kündgen and Timmons, 2010), and if $\mathrm{mad}(G)< \frac{18}{7}$ and its girth is at least 6, then $\chi_s(G)\le 5$ (Bu et al., 2009). We improve both results by showing that for a graph $G$ with $\mathrm{mad}(G)\le \frac{8}{3}$, then $\chi_s(G)\le 5$. As an immediate corollary, we obtain that a planar graph with girth at least 8 has a star 5-coloring, improving the best known girth condition for a planar graph to have a star 5-coloring (Kündgen and Timmons, 2010 and Timmons, 2008). Joint work with Ilkyoo Choi.

Yuejian Peng, Lagrangian densities of hypergraphs

Given a positive integer $n$ and an $r$-uniform hypergraph $H$, the Turán number $ex(n, H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraph on $n$ vertices. The Turán density of $H$ is defined as \[\pi(H)=\lim_{n\rightarrow\infty} { ex(n,H) \over {n \choose r } }.\] The Lagrangian density of an $r$-uniform graph $H$ is \[\pi_{\lambda}(H)=\sup \{r! \lambda(G):G\;\text{is}\;H\text{-free}\},\] where $\lambda(G)$ is the Lagrangian of $G$. The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. The Lagrangian density of an $r$-uniform hypergraph $H$ is the same as the Turán density of the extension of $H$. Therefore, these two densities of $H$ equal if every pair of vertices of $H$ is contained in an edge. For example, to determine the Lagrangian density of $K_4^{3}$ is equivalent to determine the Turán density of $K_4^{3}$. For an $r$-uniform graph $H$ on $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$, where $K_{t-1}^r$ is the complete $r$-uniform graph on $t-1$ vertices. We say that an $r$-uniform hypergraph $H$ on $t$ vertices is $\lambda$-perfect if $\pi_{\lambda}(H)= r!\lambda{(K_{t-1}^r)}$. A result of Motzkin and Straus implies that all graphs are $\lambda$-perfect. It is interesting to explore what kind of hypergraphs are $\lambda$-perfect. We present some open problems and recent results.

Zi-Xia Song, Ramsey numbers of cycles under Gallai colorings

For a graph $H$ and an integer $k\ge1$, the $k$-color Ramsey number $R_k(H)$ is the least integer $N$ such that every $k$-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $m\ge4 $ vertices. For odd cycles, Bondy and Erdős in 1973 conjectured that for all $k\ge1$ and $n\ge2$, $R_k(C_{2n+1})=n\cdot 2^k+1$. Recently, this conjecture has been verified to be true for all fixed $k$ and all $n$ sufficiently large by Jenssen and Skokan; and false for all fixed $n$ and all $k$ sufficiently large by Day and Johnson (2017). Even cycles behave rather differently in this context. Little is known about the behavior of $R_k(C_{2n})$ in general. In this talk we will present our recent results on Ramsey numbers of cycles under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles. We prove that the aforementioned conjecture holds for all $k$ and all $n$ under Gallai colorings. We also completely determine the Ramsey number of even cycles under Gallai colorings. Joint work with Yaojun Chen and Fangfang Zhang.

Tomáš Kaiser, Hamilton cycles in tough chordal graphs

Chvátal conjectured in 1973 that all graphs with sufficiently high toughness are Hamiltonian. The conjecture remains open, but it is known to be true for various classes of graphs, including chordal graphs, claw-free graphs or planar graphs. We will discuss the case of chordal graphs and outline our proof that 10-tough chordal graphs are Hamiltonian, relying on a hypergraph version of Hall’s Theorem as our main tool. This improves a previous result due to Chen et al. (1998) where the constant $10$ is replaced by $18$. Joint work with Adam Kabela.

Maho Yokota, Connectivity, toughness and forbidden subgraph conditions

Let $\textrm{conn}(G)$ and $\textrm{tough}(G)$ denote the connectivity and the toughness of $G$. We know that low connectivity implies low toughness; if $\textrm{conn}(G)\leq k$, then $\textrm{tough}(G) \leq k/2$. On the other hand, we also know the converse is not true. We can construct a graph with high connectivity and low toughness. About this, we have next proposition. Proposition 1. Let $G$ be a graph, $k$ be an integer with $k\geq 1$ and $r$ be a real number with $r>1$. If $G$ is $k$-connected and $K_{1,\lfloor r \rfloor +1}$-free, then $\textrm{tough}(G)\geq k/r$. It means high connectivity implies high toughness under the star-free condition. Our purpose is to prove assertions which can be regarded as a converse of this statement; that is say, we ask what we can say about $\mathcal H$ if high connectivity implies high toughness in the family of $\mathcal H$-free graphs. About this question, Ota and Sueiro (2013) proved the following theorem. Theorem 1 (Ota and Sueiro). Let $H$ be a connected graph and $\tau$ be a real number with $0<\tau\leq 1/2$. Almost all $H$-free connected graphs $G$ satisfy $\textrm{tough}(G)\geq \tau$ if and only if $K_{1,\lfloor 1/\tau \rfloor +1}$ contains $H$ as an induced subgraph. Our main result is high connectivity versions of this theorem. We proved the following theorems. Theorem 2. Let $H$ be a connected graph, $k$ be an integer with $k\geq 1$ and $r$ be a real number with $r>1$. Almost all $H$-free $k$-connected graphs $G$ satisfy $\textrm{tough}(G)\geq k/r$ if and only if $K_{1,\lfloor 1/\tau \rfloor +1}$ contains $H$ as an induced subgraph. Theorem 3. Let $\mathcal H=\{H_1,H_2\}$ be a family of connected graphs, $k$ be an integer with $k\geq 1$ and $r$ be a real number with $r>1$. Almost all $\mathcal H$-free $k$-connected graphs $G$ satisfy $\textrm{tough}(G)\geq k/r$ if and only if $K_{1,\lfloor 1/\tau \rfloor +1}$ contains one of $H$ as an induced subgraph.

Xuding Zhu, List colouring and Alon-Tarsi number of planar graphs

A $d$-defective colouring of a graph $G$ is a colouring of the vertices of $G$ such that each vertex $v$ has at most $d$ neighbours coloured the same colour as $v$. We say $G$ is $d$-defective $k$-choosable if for any $k$-assignment $L$ of $G$, there exists a $d$-defective $L$-colouring, i.e., a $d$-defective colouring $f$ with $f(v) \in L(v)$ for each vertex $v$. It was proved by Eaton and Hull (1999) and Škrekovski (1999) that every planar graph is $2$-defective $3$-choosable, and proved by Cushing and Kierstead (2010) that every planar graph is $1$-defective $4$-choosable. In other words, for a planar graph $G$, for any $3$-assigment $L$ of $G$, there is a subgraph $H$ with $\Delta(H) \le 2$ such that $G-E(H)$ is $L$-colourable; and for any $4$-list assignment $L$ of $G$, there is a subgraph $H$ with $\Delta(H) \le 1$ such that $G-E(H)$ is $L$-colourable. An interesting problem is whether there is a subgraph $H$ with $\Delta(H) \le 2$ such that $G-E(H)$ is $3$-choosable, and whether there is a subgraph $H$ with $\Delta(H) \le 1$ such that $G-E(H)$ is $4$-choosable. It turns out that the answer to the first question is negative and the answer to the second question is positive. Kim, Kim and I proved that there is a planar graph $G$ such that for any subgraph $H$ with $\Delta(H)=3$, $G-E(H)$ is not $3$-choosable. Grytczuk and I proved that every planar graph $G$ has a matching $M$ such that $G-M$ has Alon-Tarsi number at most $4$, and hence is $4$-choosable. The late result also implies that every planar graph is $1$-defective $4$-paintable. For a subset $X$ of $V(G)$, let $f_X$ be the function defined as $f_X(v)=4$ for $v \in X$ and $f_X(v)=5$ for $v \in V(G)-X$. Our proof also shows that every planar graph $G$ has a subset $X$ of size $|X| \ge |V(G)|/2$ such that $G$ is $f_X$-AT, which implies that $G$ is $f_X$-choosable and also $f_X$-paintable. In this talk, we shall present the proof and discuss possible strengthening of this result.