The 3rd KIAS Combinatorics Workshop

Korea Institute for Advanced Study, Seoul, Korea March 6 - 8, 2014

Information Title: The 3rd KIAS Combinatorics Workshop Date: March 6 - 8 (Thu-Sat), 2014 Venue: Room 1503, KIAS Web: http://home.kias.re.kr/MKG/h/Combinatorics/ Organizers: Jeong Han Kim (KIAS) Seog-Jin Kim (Konkuk University) Young Soo Kwon (Yeungnam University) Sang June Lee (Duksung Women’s University) Boram Park (NIMS) Seunghyun Seo (Kangwon National University)

Invited Speakers Min Chen (Zhejiang Normal University) Kwang Ju Choi (National Institute for Mathematical Sciences) Soohak Choi (Sogang University) Shinya Fujita (Yokohama City University) Danjun Huang (Zhejiang Normal University) JiSun Huh (Yonsei University) Jeong Han Kim (Korea Institute for Advanced Study) Sang-il Oum (Korea Advanced Institute of Science and Technology) Boram Park (National Institute for Mathematical Sciences) Qiaojun Shu (Soochow University/Zhejiang Normal University) Yunfang Tang (Tongji University/Zhejiang Normal University)) Weifan Wang (Zhejiang Normal University) Yingqian Wang (Zhejiang Normal University) Douglas B. West (Zhejiang Normal University/University of Illinois) Xuding Zhu (Zhejiang Normal University)

March 6, 14:10 ∼ 15:00, Talk 1

Xuding Zhu

Speaker: Xuding Zhu Affiliation: Zhejiang Normal University, China Title: Circular colouring of graphs (Part I) Abstract Graph colouring is an active topic in graph theory, which enjoys many practical applications as well as theoretical challenges. One of the many applicatios of graph colouring is to model scheduling problems. Circular colouring of graphs is a variation of graph colouring, which provides an ideal model for some periodical scheduling problem. The circular chromatic number of a graph is a refinement of the chromatic number, which has been studied extensively in the past two decades. In this talk, I shall survey results and problems in this area, as well as tools developed in the study of circular colouring of graphs. In the 2nd talk, I shall discuss in details on how to construct special graphs (such as planar graphs, graphs of large girth, line graphs, total graphs, etc.) of given circular chromatic numbers.

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March 6, 15:20 ∼ 16:10, Talk 2

Dougla B. West

Speaker: Douglas B. West Affiliation: Zhejiang Normal University/University of Illinois Title: An Introduction to the Discharging Method and List Coloring (Part I) Abstract The Discharging Method has been used in graph theory for more than 100 years. Its most famous application is the proof of the Four Color Theorem, stating that graphs embeddable in the plane have chromatic number at most 4. Despite the simplicity of the idea, discharging proofs remain mysterious to many graph theorists. We will describe how to find the “discharging rules” to create such proofs, illustrated by examples from a variety of problems related to coloring of graphs. Discharging is a tool in a two-pronged approach to inductive proofs. For example, it is often used to show that a global bound (less than b) on the average vertex degree forces the occurrence of some sparse local subgraph. Each vertex can be given a “charge” equal to its degree. Absence of the specified local configurations allows charge to be moved (via “discharging rules”) so that the final charge at each vertex is at least b, violating the hypothesized global bound. In an application of the resulting structure theorem, one shows that each such local configuration cannot occur in a minimal counterexample to the desired conclusion. This leads to the phrase “an unavoidable set of reducible configurations” to describe the overall process. List coloring is a modern topic in coloring theory where the discharging method has many applications. When resources are not always available at all vertices, each vertex v has a “list” L(v) of colors available to be used on it. An L-coloring is a proper coloring with the color on each vertex chosen from its list. The list chromatic number is the least k such that the graph has an L-coloring whenever each list has size k; this is always at least the ordinary chromatic number. We seek bounds on the list chromatic number in various settings, some proved by the discharging method.

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March 6, 16:40 ∼ 15:10, Talk 3

Shinya Fujita

Speaker: Shinya Fujita Affiliation: Yokohama City University, Japan Title: Covering vertices by monochromatic components in edge-colored graphs and hypergraphs Abstract In this talk, I would like to introduce some recent results on covering vertices by monochromatic components in edge-colored graphs and hypergraphs. The aim of this topic is to find monochromatic components covering the whole vertex set of an edge-colored (hyper)graph. We will mainly consider such problems for the following edge-colored graphs G: i) G is a compete graph ii) G is a complete bipartite graph iii) G is a (hyper) graph with given independence number

References [1] G. Chen, S. Fujita, A. Gyarfas, J. Lehel, A. Toth, Around a biclique cover conjecture, Preprint- arXiv:1212.6861v1 [math.CO] [2] S. Fujita, M. Furuya, A. Gyarfas, A. Toth, Partition of graphs and hypergraphs into monochromatic connected parts, Electronic Journal of Combinatorics, Vol. 19(3), 2012 [3] S. Fujita, C. Magnant, Forbidden rainbow subgraphs that force large highly connected monochromatic subgraphs, SIAM Journal on Discrete Mathematics, Vol. 27(3), 2013

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March 6, 17:20 ∼ 17:50, Talk 4

Boram Park

Speaker: Boram Park Affiliation: National Institute for Mathematical Sciences, Korea Title: Coloring the square of Kneser graph K(2k + r, k) for 1 ≤ r ≤ k − 1 Abstract The Kneser graph K(n, k) is the graph whose vertices are the k-elements subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The square G2 of a graph G is the graph defined on V (G) such that two vertices u and v are adjacent in G2 if the distance between u and v is at most 2 in G. The problem of computing χ(K 2 (n, k)), which was originally posed by F¨uredi, was introduced and discussed in [2]. Determining the chromatic number of the square of the Kneser graph K(n, k) is an interesting graph coloring problem, and also it is related with intersecting family and constant weight code. The best known upper bound for χ(K 2 (2k + 1, k)) was obtained in [3]. Kim and Park [3] showed that for k ≥ 2, (a) χ(K 2 (2k + 1, k)) ≤ 38 k + 20 3 2 (b) χ(K (2k + 1, k)) ≤ 2k + 2 if 2k + 1 = 2t − 1 for some integer t, )r for any 2 ≤ r ≤ k − 1. (c) χ(K 2 (2k + r, k)) ≤ (r + 2)(3k + 3r+3 2 In this talk, we will present recent new results on the chromatic number of the square of the Kneser graph, which improve and generalize the results in [3]. This is joint work with Seog-Jin Kim.

References [1] J.-Y. Chen, K.-W. Lih, and J. Wu: Coloring the square of the Kneser graph KG(2k + 1, k) and the Schrijver graph SG(2k + 2, 2), Discrete Appl. Math. 157 (2009), 170–176. [2] S.-J. Kim and K. Nakprasit: On the chromatic number of the square of the Kneser graph K(2k + 1, k), Graph. Combinator. 20 (2004), 79–90. [3] S.-J. Kim and B. Park: Improved bounds on the chromatic numbers of the square of Kneser graphs. Discrete Math., 315 (2014), 69-74.

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March 8, 9:30 ∼ 10:20, Talk 5

Dougla B. West

Speaker: Douglas B. West Affiliation: Zhejiang Normal University/University of Illinois Title: An Introduction to the Discharging Method and List Coloring (Part II) Abstract The Discharging Method has been used in graph theory for more than 100 years. Its most famous application is the proof of the Four Color Theorem, stating that graphs embeddable in the plane have chromatic number at most 4. Despite the simplicity of the idea, discharging proofs remain mysterious to many graph theorists. We will describe how to find the “discharging rules” to create such proofs, illustrated by examples from a variety of problems related to coloring of graphs. Discharging is a tool in a two-pronged approach to inductive proofs. For example, it is often used to show that a global bound (less than b) on the average vertex degree forces the occurrence of some sparse local subgraph. Each vertex can be given a “charge” equal to its degree. Absence of the specified local configurations allows charge to be moved (via “discharging rules”) so that the final charge at each vertex is at least b, violating the hypothesized global bound. In an application of the resulting structure theorem, one shows that each such local configuration cannot occur in a minimal counterexample to the desired conclusion. This leads to the phrase “an unavoidable set of reducible configurations” to describe the overall process. List coloring is a modern topic in coloring theory where the discharging method has many applications. When resources are not always available at all vertices, each vertex v has a “list” L(v) of colors available to be used on it. An L-coloring is a proper coloring with the color on each vertex chosen from its list. The list chromatic number is the least k such that the graph has an L-coloring whenever each list has size k; this is always at least the ordinary chromatic number. We seek bounds on the list chromatic number in various settings, some proved by the discharging method.

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March 7, 10:50 ∼ 11:20, Talk 6

Min Chen

Speaker: Min Chen Affiliation: Zhejiang Normal University, China Title: Acyclic list coloring of planar graphs Abstract Let G = (V, E) be a graph. A proper vertex coloring of G is acyclic if G contains no bicolored cycle. Namely, every cycle of G must be colored with at least three colors. G is acyclically Llist colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper acyclic coloring π of G such that π(v) ∈ L(v) for all v ∈ V . If G is acyclically L-list colorable for any list assignment with |L(v)| ≥ k for all v ∈ V , then G is acyclically k-choosable. In 1976, Steinberg conjectured that every planar graph without 4- and 5-cycles is 3-colorable. This conjecture cannot be improved to 3-choosable basing on the examples given by Voigt and independently, by Montassier. In this talk, We will show that planar graphs without 4- and 5cycles are acyclically 4-choosable. This result is also a new approach to the conjecture proposed by Montassier, Raspaud and Wang, which says that every planar graph without 4-cycles is acyclically 4-choosable.

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March 7, 11:30 ∼ 12:00, Talk 7

Danjun Huang

Speaker: Danjun Huang Affiliation: Zhejiang Normal University, China Title: The adjacent vertex distinguishing index of planar graphs Abstract The adjacent vertex distinguishing index of a graph G, denoted by χ0a (G), is the minimum number of colors assigned properly to the edges of G such that any pair of adjacent vertices have distinct sets of colors. In this talk, we show that if G is a connected planar graph with maximum degree ∆ ≥ 12, then χ0a (G) ≤ ∆ + 2. This is a joint work with M. Horˇna´ k and W. Wang.

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March 7, 14:00 ∼ 14:50, Talk 8

Jeong Han Kim

Speaker: Jeong Han Kim Affiliation: Korea Institute for Advanced Study, Korea Title: Two Approaches to Sidorenko’s Conjecture Abstract Sidorenko’s conjecture states that for every bipartite graph H on {1, . . . , k} Z

Y

|V (H)|

h(xi , yj )dµ

Z ≥

h(x, y) dµ

2

|E(H)|

(i,j)∈E(H)

holds, where µ is the Lebesgue measure on [0, 1] and h is a bounded, non-negative, symmetric, measurable function on [0, 1]2 . An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph H to a graph G is asymptotically at least the expected number of homomorphisms from H to the Erd˝os-R´enyi random graph with the same expected edge density as G. In this paper, we present two approaches to the conjecture. First, we introduce the notion of tree-arrangeability, where a bipartite graph H with bipartition A ∪ B is tree-arrangeable if neighborhoods of vertices in A have a certain tree-like structure. We show that Sidorenko’s conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko’s conjecture holds if there are two vertices a1 , a2 in A such that each vertex a ∈ A satisfies N (a) ⊆ N (a1 ) or N (a) ⊆ N (a2 ), and also implies a recent result of Conlon, Fox, and Sudakov. Second, if T is a tree and H is a bipartite graph satisfying Sidorenko’s conjecture, then it is shown that the Cartesian product T  H of T and H also satisfies Sidorenko’s conjecture. This result implies that, for all d ≥ 2, the d-dimensional grid with arbitrary side lengths satisfies Sidorenko’s conjecture. This is joint work with Choongbum Lee (MIT) and Joonkyung Lee (U. of Oxford)

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March 7, 15:10 ∼ 16:00, Talk 9

Weifan Wang

Speaker: Weifan Wang Affiliation: Zhejiang Normal University, China Title: Simultaneous Colorings of Plane Graphs Abstract Given a plane graph G = (V, E, F ), we can define the vertex coloring for V , the edge coloring for E, the face coloring for F, the total coloring for V ∪E, the coupled coloring for V ∪F , the edgeface coloring for E ∪F , and the entire coloring V ∪E ∪F , such that adjacent or incident elements have different colors. In this talk we shall give a survey on these colorings and list colorings of plane graphs. Some recent results and open problems about this direction are mentioned.

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March 7, 16:30 ∼ 17:00, Talk 10

Yingqian Wang

Speaker: Yingqian Wang Affiliation: Zhejiang Normal University, China Title: Coloring planar graphs with three colors Abstract Let d1 , d2 , . . . , dk be k non-negative integers. A (d1 , d2 , . . . , dk )-coloring of a graph G = (V ; E) is a partition (V1 , V2 , . . . , Vk )of the vertex set V of G such that the subgraph G[Vi ] has maximum degree at most di for i = 1, 2, . . . , k. This notion is clearly a generalization of the classical proper vertex coloring. In terms of this notion, every planar graph is (0, 0, 0, 0)-colorable. A central conjecture on 3-colorability of planar graphs, proposed by Steinberg in 1976, asserts that every planar graph without cycles of length 4 or 5 is (0, 0, 0)-colorable. More generally, what is the set of integers, say I, for every i ∈ I, if any, every planar graphs without cycles of length 4 or i is (0, 0, 0)-colorable? A natural approach to attack this general coloring problem seems to study various (d1 , d2 , d3 )-colorability of planar graphs without cycles of length 4 or i (i ≥ 5). In this talk, we rst give a survey on various (d1 , d2 , d3 )-coloring of planar graphs with some forbidden short cycles; then give an outline of the proof of a result obtained recently which asserts that every planar graph without 5-cycles is (1, 1, 1)-colorable, namely, every such graph can be decomposed into a matching and a 3-colorable graph; finally give some problems in this direction for further study.

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March 7, 17:10 ∼ 17:40, Talk 11

Kwang Ju Choi

Speaker: Kwang Ju Choi Affiliation: National Institute for Mathematical Sciences, Korea Title: The spindle surface and nearly planar graphs Abstract The spindle surface is the pinched surface formed by identifying two points on the sphere. When we investigate this pseudo-surface, we notice that the class of embeddable graphs on the spindle surface contains the class of nearly planar graphs, that is, graphs that are edgeless or have an edge whose deletion results in a planar graph. We show that all but finitely many graphs that are not nearly planar and do not contain one particular graph have a well-understood structure based on large M¨obius ladders.

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March 8, 9:30 ∼ 10:20, Talk 12

Xuding Zhu

Speaker: Xuding Zhu Affiliation: Zhejiang Normal University, China Title: Circular colouring of graphs (Part II) Abstract Graph colouring is an active topic in graph theory, which enjoys many practical applications as well as theoretical challenges. One of the many applicatios of graph colouring is to model scheduling problems. Circular colouring of graphs is a variation of graph colouring, which provides an ideal model for some periodical scheduling problem. The circular chromatic number of a graph is a refinement of the chromatic number, which has been studied extensively in the past two decades. In this talk, I shall survey results and problems in this area, as well as tools developed in the study of circular colouring of graphs. In the 2nd talk, I shall discuss in details on how to construct special graphs (such as planar graphs, graphs of large girth, line graphs, total graphs, etc.) of given circular chromatic numbers.

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March 8, 10:50 ∼ 11:20, Talk 13

Soohak Choi

Speaker: Soohak Choi Affiliation: Sogang University, Korea Title: Local duality theorem for q-ary 1-perfect codes Abstract Let u be a vector in Fqn with u = (u1 , u2 , . . . , un ). The support s(u) of u is the set of nonzero coordinate positions of u. Let I be a subset of [n] , {1, 2, . . . , n}. The I-face BIw (centered at w) is defined as BIw = {v + w : s(v) ⊆ I}. The local weight enumerator of a code C is defined by X WCI,w (x, y) = x|I|−|s(c−w)| y |s(c−w)| . c∈C∩BIw

This is introduced by Vasileva [2], which generalize the notion of a weight enumerator. For an arbitrary binary 1-perfect code, the connection between local weight enumerators of a code on a pair of orthogonal faces of dimension k and (nk) was investigated by Vasileva [2]. The more explicit formula was derived by Hyun [1]. In this talk, we derive the relationship between local weight enumerator of q-ary 1-perfect code in a face and that in the orthogonal face. As an application of our result, we compute the local weight enumerators of a shortened, doubly-shortened, and triply-shortened q-ary 1-perfect code. This is joint work with H. K. Kim and J. Y. Hyun.

References [1] J. Y. Hyun, Generalized MacWilliams identities and their applications to perfect binary codes. Des. Codes Cryptogr. 50(2) (2009) 173–185. [2] A. Y. Vasil’eva, Local spectra of perfect binary codes, Russian translations. II, Discrete Appl. Math. 135 (2004), 301–307.

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March 8, 11:30 ∼ 12:00, Talk 14

JiSun Huh

Speaker: JiSun Huh Affiliation: Yonsei University, Korea Title: The Chung-Feller Theorem on Generalized Lattice Paths Abstract We study bicolored lattice paths whose peaks and valleys are colored in black only and other vertices either in black or in white. After enumerating these paths we find a natural bijection between the bicolored paths and generalized lattice paths with arbitrary step sizes. We also consider generalized paths with flaws and find a Chung-Feller theorem for them by combinatorial methods.

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March 8, 14:00 ∼ 14:50, Talk 15

Sang-il Oum

Speaker: Sang-il Oum Affiliation: Korea Advanced Institute of Science and Technology, Korea Title: Unifying duality theorems for width parameters Abstract We provide a framework that generalizes the tree-width bramble duality theorem as well as branch-width tangle duality theorem, based on orientations of separations. This is joint work with Reinhard Diestel.

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March 8, 15:20 ∼ 15:50, Talk 16

Qiaojun Shu

Speaker: Qiaojun Shu Affiliation: Soochow University/Zhejiang Normal University, China Title: Acyclic Edge Coloring of Graphs Abstract An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a0 (G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. It was conjectured that a0 (G) ≤ ∆ + 2 for any simple graph G with maximum degree ∆. In this talk, we first give a survey on the acyclic edge coloring and acyclic list edge coloring of graphs. Then we state several results we have obtained recently on this direction. In particular, we show that every planar graph G with maximum degree ∆ has a0 (G) ≤ ∆ + 7, which improves a known result that a0 (G) ≤ ∆ + 12 by Basavaraju et al. We also show that all 4-regular graphs are acyclically edge 6-colorable. This is joint work with Weifan Wang.

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March 8, 16:00 ∼ 16:30, Talk 17

Yunfang Tang

Speaker: Yunfang Tang Affiliation: Tongji University/Zhejiang Normal University, China Title: Total weight choosability of the k-Cone graphs Abstract A total weighting of a graph G is a mapping φ that assigns a weightP to each vertex and each edge of G. The vertex-sum of v ∈ V (G) with respect to φ is Sφ (v) = e∈E(v) φ(e) + φ(v). A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph G = (V, E) is called (k, k 0 )-choosable if the following is true: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k 0 real numbers, there is a proper total weighting φ with φ(y) ∈ L(y) for any y ∈ V ∪ E. In this paper, we prove that for any graph G 6= K1 , the k-cone graph of G is (1, 4)-choosable. Especially, we show that the mycielski graph of G is (1, 4)-choosable. This is joint work with Xuding Zhu.

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