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BEGIN:VEVENT
DTSTART;VALUE=DATE:20191026
DTEND;VALUE=DATE:20191028
DTSTAMP:20260424T044601
CREATED:20190910T133412Z
LAST-MODIFIED:20240707T090017Z
UID:1366-1572048000-1572220799@dimag.ibs.re.kr
SUMMARY:Extremal and Structural Graph Theory (2019 KMS Annual Meeting)
DESCRIPTION:Focus Session @ 2019 KMS Annual Meeting\nA focus session “Extremal and Structural Graph Theory” at the 2019 KMS Annual Meeting is organized by Sang-il Oum. URL: http://www.kms.or.kr/meetings/fall2019/ \nSpeakers\n\nIlkyoo Choi (최일규)\, Hankuk University of Foreign Studies\nKevin Hendrey\, IBS Discrete Mathematics Group\nPascal Gollin\, IBS Discrete Mathematics Group\nJaehoon Kim (김재훈)\, KAIST\nRingi Kim (김린기)\, KAIST\nSeog-Jin Kim (김석진)\, Konkuk University\nO-joung Kwon (권오정)\, Incheon National University / IBS Discrete Mathematics Group\nZi-Xia Song (宋梓霞)\, University of Central Florida\nCasey Tompkins\, IBS Discrete Mathematics Group\n\nSchedules\n\nSlot A: October 26\, 2019 Saturday\, 9:00am-10:15am\n\nSeog-Jin Kim\nKevin Hendrey\nIlkyoo Choi\n\n\nSlot B: October 26\, 2019 Saturday\, 10:40am-11:55pm\n\nJaehoon Kim\nRingi Kim\nCasey Tompkins\n\n\nSlot D: October 27\, 2019 Sunday\, 9:00am-10:15am\n\nZi-Xia Song\nPascal Gollin\nO-joung Kwon\n\n\n\nAbstracts\nIlkyoo Choi (최일규)\, Degeneracy and colorings of squares of planar graphs without 4-cycles\nWe prove several results on coloring squares of planar graphs without 4-cycles. First\, we show that if $G$ is such a graph\, then $G^2$ is $(\Delta(G)+72)$-degenerate. This implies an upper bound of $\Delta(G)+73$ on the chromatic number of $G^2$ as well as on several variants of the chromatic number such as the list-chromatic number\, paint number\, Alon–Tarsi number\, and correspondence chromatic number. We also show that if $\Delta(G)$ is sufficiently large\, then the upper bounds on each of these parameters of $G^2$ can all be lowered to $\Delta(G)+2$ (which is best possible). To complement these results\, we show that 4-cycles are unique in having this property. Specifically\, let $S$ be a finite list of positive integers\, with $4\notin S$. For each constant $C$\, we construct a planar graph $G_{S\,C}$ with no cycle with length in $S$\, but for which $\chi(G_{S\,C}^2) > \Delta(G_{S\,C})+C$. This is joint work with Dan Cranston and Theo Pierron. \nKevin Hendrey\, Defective and clustered choosability of sparse graphs\nAn (improper) graph colouring has defect $d$ if each monochromatic subgraph has maximum degree at most $d$\, and has clustering $c$ if each monochromatic component has at most $c$ vertices. Given an infinite class of graphs $\mathcal{G}$\, it is interesting to ask for the minimum integer $\chi_{\Delta}(\mathcal{G})$ for which there exist an integer $d$ such that every graph in $\mathcal{G}$ has a $\chi_{\Delta}(\mathcal{G})$-colouring with defect at most $d$\, and the minimum integer $\chi_{\star}(\mathcal{G})$-colouring for which there exist an integer $c$ such that every graph in $\mathcal{G}$ has a $\chi_{\star}(\mathcal{G})$-colouring with clustering at most $c$. We explore clustered and defective colouring in graph classes with bounded maximum average degree. As an example\, our results show that every earth-moon graph has an 8-colouring with clustering at most 405. Our results hold in the stronger setting of list-colouring. Joint work with David Wood. \nPascal Gollin\, Progress on the Ubiquity Conjecture\nA classic result of Halin says that if a graph $G$ contains for every $n \in \mathbb{N}$ as subgraphs $n$ disjoint rays\, i.e.\, one-way infinite paths\, then $G$ already contains infinitely many disjoint rays as subgraphs. We say a graph $G$ is ubiquitous with respect to the subgraph relation if whenever a graph $\Gamma$ contains $n$ disjoint copies of $G$ as a subgraph for all $n \in \mathbb{N}$\, then $\Gamma$ already contains infinitely many disjoint copies of $G$ as a subgraph. We define ubiquity w.r.t. the minor relation or topological minor relation analogously. A fundamental conjecture about infinite graphs due to Andreae is the Ubiquity Conjecture. It states that every locally finite connected graph is ubiquitous w.r.t. the minor relation. In a series of papers we make progress on the conjecture proving various ubiquity results w.r.t. both the topological minor relation and the minor relation\, making use of well-quasi ordering techniques. \nJaehoon Kim (김재훈)\, A rainbow version of Dirac’s theorem\nFor a collection $\mathbf{G}=\{G_1\,\dots\, G_s\}$ of graphs\, we say that a graph $H$ is $\mathbf{G}$-transversal\, or $\mathbf{G}$-rainbow\, if there exists a bijection $\phi:E(H)\rightarrow [s]$ with $e\in G_{\phi(e)}$ for all $e\in E(H)$. Aharoni conjectured that if for each $i\in [r]$\, then graph $G_i$ is an $n$-vertex graph on the same vertex set $V$ and $\delta(G_i)\geq n/2$ for all $i\in [s]$\, then there exists a $\mathbf{G}$-transversal Hamilton cycle on $V$. We prove this conjecture. We also prove a similar result for $K_r$-factors. This is joint work with Felix Joos. \nRingi Kim (김린기)\, Obstructions for partitioning into forests\nFor a class $\mathcal{C}$ of graphs\, we define $\mathcal{C}$-edge-brittleness of a graph $G$ as the minimum $\ell$ such that the vertex set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal{C}$ and there are $\ell$ edges having ends in distinct parts. In this talk\, we characterize classes of graphs having bounded $\mathcal{C}$-edge-brittleness for a class $\mathcal{C}$ of forests in terms of forbidden obstructions. This is joint work with Sang-il Oum and Sergey Norin. \nSeog-Jin Kim (김석진)\, The Alon-Tarsi number of subgraphs of a planar graph\nThis paper constructs a planar graph $G_1$ such that for any subgraph $H$ of $G_1$ with maximum degree $\Delta(H) \le 3$\, $G_1-E(H)$ is not $3$-choosable\, and a planar graph $G_2$ such that for any star forest $F$ in $G_2$\, $G_2-E(F)$ contains a copy of $K_4$ and hence $G_2-E(F)$ is not $3$-colourable. On the other hand\, we prove that every planar graph $G$ contains a forest $F$ such that the Alon-Tarsi number of $G – E(F)$ is at most $3$\, and hence $G – E(F)$ is 3-paintable and 3-choosable. This is joint work with Ringi Kim and Xuding Zhu. \nO-joung Kwon (권오정)\, Erdős-Pósa property of H-induced subdivisions\nA class $\mathcal{F}$ of graphs has the induced Erdős-Pósa property if there exists a function $f$ such that for every graph $G$ and every positive integer $k$\, $G$ contains either $k$ pairwise vertex-disjoint induced subgraphs that belong to $\mathcal{F}$\, or a vertex set of size at most $f(k)$ hitting all induced copies of graphs in $\mathcal{F}$. Kim and Kwon (SODA’18) showed that for a cycle $C_{\ell}$ of length $\ell$\, the class of $C_{\ell}$-subdivisions has the induced Erdős-Pósa property if and only if $\ell\le 4$. In this paper\, we investigate whether or not the class of $H$-subdivisions has the induced Erdős-Pósa property for other graphs $H$. We completely settle the case when $H$ is a forest or a complete bipartite graph. Regarding the general case\, we identify necessary conditions on $H$ for the class of $H$-subdivisions to have the induced Erdős-Pósa property. For this\, we provide three basic constructions that are useful to prove that the class of the subdivisions of a graph does not have the induced Erdős-Pósa property. Among remaining graphs\, we prove that if $H$ is either the diamond\, the $1$-pan\, or the $2$-pan\, then the class of $H$-subdivisions has the induced Erdős-Pósa property. \nZi-Xia Song\, On the size of $(K_t\,\mathcal{T}_k)$-co-critical graphs\nGiven an integer $r\ge1$ and graphs $G\, H_1\, \ldots\, H_r$\, we write $G \rightarrow ({H}_1\, \ldots\, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1\, \ldots\, r\}$. A non-complete graph $G$ is $(H_1\, \ldots\, H_r)$-co-critical if $G \nrightarrow ({H}_1\, \ldots\, {H}_r)$\, but $G+e\rightarrow ({H}_1\, \ldots\, {H}_r)$ for every edge $e$ in $\overline{G}$. Motivated by Hanson and Toft’s conjecture\, we study the minimum number of edges over all $(K_t\, \mathcal{T}_k)$-co-critical graphs on $n$ vertices\, where $\mathcal{T}_k$ denotes the family of all trees on $k$ vertices. In this talk\, we will present our recent results on this topic. This is joint work with Jingmei Zhang. \nCasey Tompkins\, Generalizations of the Erdős-Gallai theorem\nA classical result of Erdős and Gallai bounds the number of edges in an $n$-vertex graph with no path of length $k$ (denoted $P_k$). In this talk\, I will discuss generalizations of this result in multiple settings. In one direction\, I will consider so-called generalized Turán problems for $P_k$-free graphs. That is\, rather than maximizing the number of edges\, we consider maximizing the number of copies of some graph $H$. Building on results of Luo where $H$ is a clique\, we consider the case when $H$ is also a path. In another direction\, I will consider Erdős-Gallai type problems in a hypergraph setting. Here I will discuss recent results involving forbidding Berge copies of a path\, including the solution to some problems and conjectures of Győri\, Katona and Lemons as well Füredi\, Kostochka and Luo. I will also mention some Kopylov-type variants of this problem where the hypergraph is assumed to be connected. Moreover\, I will discuss some recent work on forbidding Berge copies of a tree. Finally\, as time permits I will mention some colored variants of the Erdős-Gallai problem. The new results presented are joint work with various subsets of the authors Akbar Davoodi\, Beka Ergemlidze\, Ervin Győri\, Abhishek Methuku\, Nika Salia\, Mate Vizer\, Oscar Zamora.
URL:https://dimag.ibs.re.kr/event/2019-10-26/
LOCATION:Room 426\, Hong-Mun Hall\, Hongik University\, Seoul
CATEGORIES:Workshops and Conferences
END:VEVENT
BEGIN:VEVENT
DTSTART;VALUE=DATE:20191031
DTEND;VALUE=DATE:20191105
DTSTAMP:20260424T044601
CREATED:20190514T145906Z
LAST-MODIFIED:20240707T090002Z
UID:863-1572480000-1572911999@dimag.ibs.re.kr
SUMMARY:The 2nd East Asia Workshop on Extremal and Structural Graph Theory
DESCRIPTION: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. \nDate\nOct 31\, 2019 (Arrival Day) – Nov 4\, 2019 (Departure Day) \nVenue and Date\n1st floor  Diamond Hall \nUTOP 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. \nThe UTOP UBLESS Hotel is located at the beautiful Hamdeok Beach of the Jeju Island.\nInvited Speakers\n\nPing Hu\, Sun Yat-Sen University\, China\nJaehoon Kim\, KAIST\, Korea\nO-joung Kwon\, Incheon National University and IBS Discrete Mathematics Group\, Korea\nJoonkyung Lee\, University of Hamburg\, Germany\nBinlong Li\, Northwestern Polytechnical University\, China\nHongliang Lu\, Xi’an Jiaotong University\, China\nAbhishek Methuku\, IBS Discrete Mathematics Group\, Korea\nAtsuhiro Nakamoto\, Yokohama National University\, Japan\nKenta Noguchi\, Tokyo University of Science\, Japan\nKenta Ozeki\, Yokohama National University\, Japan\nBoram Park\, Ajou University\, Korea\nYuejian Peng\, Hunan University\, China\nZi-Xia Song\, University of Central Florida\, U.S.A.\nTomáš Kaiser\, University of West Bohemia\, Czech Republic.\nMaho Yokota\, Tokyo University of Science\, Japan.\nXuding Zhu\, Zhejiang Normal University\, China\n\nMore speakers to be announced as soon as confirmed. Last update: September 10. \nProgram\nDay 0 (Oct. 31 Thursday)\n\n4:00PM-6:00Pm Registration and Discussions\n\nDay 1 (Nov. 1 Friday)\n\n9:00AM-9:20AM Opening address\n9:20AM-9:50AM Jaehoon Kim\, A quantitative result on the polynomial Schur’s theorem\n10:00AM-10:30AM Yuejian Peng\, Lagrangian densities of hypergraphs\n10:30AM-10:50AM Coffee Break\n10:50AM-11:20AM Atsuhiro Nakamoto\, Geometric quadrangulations on the plane\n11:30AM-12:00PM Ping Hu\, The inducibility of oriented stars\n2:00PM-2:30PM Boram Park\, 5-star coloring of some sparse graphs\n2:40PM-3:10PM Kenta Ozeki\, An orientation of graphs with out-degree constraint\n3:10PM-3:30PM Coffee Break\n3:30PM-5:30PM Problem session\n\nDay 2 (Nov. 2 Saturday)\n\n9:20AM-9:50AM Xuding Zhu\, List colouring and Alon-Tarsi number of planar graphs\n10:00AM-10:30AM O-joung Kwon\, A survey of recent progress on Erdős-Pósa type problems\n10:30AM-10:50AM Coffee Break\n10:50AM-11:20AM Kenta Noguchi\, Extension of a quadrangulation to triangulations\, and spanning quadrangulations of a triangulation\n11:30AM-12:00PM Zi-Xia Song\, Ramsey numbers of cycles under Gallai colorings\n2:00PM-2:30PM Binlong Li\, Cycles through all finite vertex sets in infinite graphs\n2:40PM-3:10PM Tomáš Kaiser\, Hamilton cycles in tough chordal graphs\n3:20PM-3:50PM Abhishek Methuku\, On a hypergraph bipartite Turán problem\n3:50PM-4:10PM Coffee Break\n4:10PM-6:00PM Problem session and discussion\n\nDay 3 (Nov. 3 Sunday)\n\n9:20AM-9:50AM Joonkyung Lee\, Odd cycles in subgraphs of sparse pseudorandom graphs\n10:00AM-10:30AM Maho Yokota\, Connectivity\, toughness and forbidden subgraph conditions\n10:30AM-10:50AM Coffee Break\n10:50AM-11:20AM Hongliang Lu\, On minimum degree thresholds for fractional perfect matchings and near perfect matchings in hypergraphs\n11:30AM-12:00PM Contributed Talks\n2:00PM-6:00PM Problem session / Discussions / Hike\n\nDay 4 (Nov. 4 Monday)\n\n9:00AM-10:30AM Discussions\n\nHistory\n\n1st East Asia Workshop on Extremal and Structural Graph Theory\n\nNov. 30-Dec. 2\, 2018.\nHeld at and sponsored by Shanghai Center for Mathematical Sciences in China\, under the name “2018 SCMS Workshop on Extremal and Structural Graph Theory”.\nOrganizers: Ping Hu\, Seog-Jin Kim\, Kenta Ozeki\, Hehui Wu.\n\n\n\nOrganizers\n\nSeog-Jin Kim\, Konkuk University\, Korea.\nSang-il Oum\, IBS Discrete Mathematics Group\, Korea and KAIST\, Korea.\nKenta Ozeki\, Yokohama National University\, Japan.\nHehui Wu\, Shanghai Center for Mathematical Sciences\, China.\n\nSponsor\nIBS Discrete Mathematics Group\, Korea. \nAbstracts\nPing Hu\, The inducibility of oriented stars\nLet $S_{k\,\ell}$ 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. \nJaehoon Kim\, A quantitative result on the polynomial Schur’s theorem\nRecently\, 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. \nO-joung Kwon\, A survey of recent progress on Erdős-Pósa type problems\nA 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. \nJoonkyung Lee\, Odd cycles in subgraphs of sparse pseudorandom graphs\n  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. \nBinlong Li\, Cycles through all finite vertex sets in infinite graphs\nA closed curve in the Freudenthal compactification $|G|$ of an infinite locally finite graph $G$ is called a Hamiltonian curve 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. \nHongliang Lu\, On minimum degree thresholds for fractional perfect matchings and near perfect matchings in hypergraphs\nWe 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. \nAbhishek Methuku\, On a hypergraph bipartite Turán problem\nLet $t$ 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. \nAtsuhiro Nakamoto\, Geometric quadrangulations on the plane\nLet $P$ 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. \nKenta Noguchi\, Extension of a quadrangulation to triangulations\, and spanning quadrangulations of a triangulation\nA 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. \nKenta Ozeki\, An orientation of graphs with out-degree constraint\nAn 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. \nBoram Park\, 5-star coloring of some sparse graphs\nA 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. \nYuejian Peng\, Lagrangian densities of hypergraphs\nGiven 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. \nZi-Xia Song\, Ramsey numbers of cycles under Gallai colorings\nFor 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. \nTomáš Kaiser\, Hamilton cycles in tough chordal graphs\nChvátal 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. \nMaho Yokota\, Connectivity\, toughness and forbidden subgraph conditions\nLet $\textrm{conn}(G)$ 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. \nXuding Zhu\, List colouring and Alon-Tarsi number of planar graphs\nA $d$-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.
URL:https://dimag.ibs.re.kr/event/2019-east-asia-graph-theory/
LOCATION:UTOP UBLESS Hotel\, Jeju\, Korea (유탑유블레스호텔제주)
CATEGORIES:Workshops and Conferences
ORGANIZER;CN="Seog-Jin Kim (%EA%B9%80%EC%84%9D%EC%A7%84)":MAILTO:skim12@konkuk.ac.kr
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