20 research outputs found

    On List Coloring with Separation of the Complete Graph and Set System Intersections

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    We consider the following list coloring with separation problem: Given a graph GG and integers a,ba,b, find the largest integer cc such that for any list assignment LL of GG with L(v)=a|L(v)|= a for any vertex vv and L(u)L(v)c|L(u)\cap L(v)|\le c for any edge uvuv of GG, there exists an assignment φ\varphi of sets of integers to the vertices of GG such that φ(u)L(u)\varphi(u)\subset L(u) and φ(v)=b|\varphi(v)|=b for any vertex uu and φ(u)φ(v)=\varphi(u)\cap \varphi(v)=\emptyset for any edge uvuv. Such a value of cc is called the separation number of (G,a,b)(G,a,b). Using a special partition of a set of lists for which we obtain an improved version of Poincar\'e's crible, we determine the separation number of the complete graph KnK_n for some values of a,ba,b and nn, and prove bounds for the remaining values.Comment: 18 page

    EUROCOMB 21 Book of extended abstracts

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    Algorithmic approaches to problems in probabilistic combinatorics

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    The probabilistic method is one of the most powerful tools in combinatorics; it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing and analyzing algorithms to find and construct these objects in an efficient way. In this dissertation we obtain four results using algorithmic approaches in probabilistic method: 1. We study the structural properties of the triangle-free graphs generated by a semi-random variant of triangle-free process and obtain a packing extension of Kim's famous R(3,t) results. This allows us to resolve a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo, and answer a problem in extremal graph theory by Esperet, Kang, and Thomasse. 2. We determine the order of magnitude of Prague dimension, which concerns efficient encoding and decomposition of graphs, of binomial random graph with high probability. We resolve conjectures by Furedi and Kantor. Along the way, we prove a Pippenger-Spencer type edge coloring result for random hypergraphs with edges of size O(log n). 3. We analyze the number set generated by r-AP free process, which answers a problem raised by Li and has connection with van der Waerden number in additive combinatorics and Ramsey theory. 4. We study a refined alteration approach to construct H-free graphs in binomial random graphs, which has applications in Ramsey games.Ph.D

    29th International Symposium on Algorithms and Computation: ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan

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    Coloring and covering problems on graphs

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    The \emph{separation dimension} of a graph GG, written π(G)\pi(G), is the minimum number of linear orderings of V(G)V(G) such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} πf(G)\pi_f(G), which is the minimum of a/ba/b such that some aa linear orderings (repetition allowed) separate every two nonincident edges at least bb times. In contrast to separation dimension, we show fractional separation dimension is bounded: always πf(G)3\pi_f(G)\le 3, with equality if and only if GG contains K4K_4. There is no stronger bound even for bipartite graphs, since πf(Km,m)=πf(Km+1,m)=3mm+1\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}. We also compute πf(G)\pi_f(G) for cycles and some complete tripartite graphs. We show that πf(G)<2\pi_f(G)<\sqrt{2} when GG is a tree and present a sequence of trees on which the value tends to 4/34/3. We conjecture that when n=3mn=3m the K4K_4-free nn-vertex graph maximizing πf(G)\pi_f(G) is Km,m,mK_{m,m,m}. We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let π(G)\pi^\circ(G) be the number of circular orderings needed to separate all pairs, and let πf(G)\pi_f^\circ(G) be the fractional version. Among our results: (1) π(G)=1\pi^\circ(G)=1 if and only GG is outerplanar. (2) π(G)2\pi^\circ(G)\le2 when GG is bipartite. (3) π(Kn)log2log3(n1)\pi^\circ(K_n)\ge\log_2\log_3(n-1). (4) πf(G)32\pi_f^\circ(G)\le\frac{3}{2}, with equality if and only if K4GK_4\subseteq G. (5) πf(Km,m)=3m32m1\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}. A \emph{star kk-coloring} is a proper kk-coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to prove that every graph with maximum average degree less than 52\frac{5}{2} has an \Ifp, which is sharp and improves the result of Bu, Cranston, Montassier, Raspaud, and Wang (2009). As a corollary, we gain that every planar graph with girth at least 10 has a star 4-coloring. A proper vertex coloring of a graph GG is \emph{rr-dynamic} if for each vV(G)v\in V(G), at least min{r,d(v)}\min\{r,d(v)\} colors appear in NG(v)N_G(v). We investigate 33-dynamic versions of coloring and list coloring. We prove that planar and toroidal graphs are 3-dynamically 10-choosable, and this bound is sharp for toroidal graphs. Given a proper total kk-coloring cc of a graph GG, we define the \emph{sum value} of a vertex vv to be c(v)+uvE(G)c(uv)c(v) + \sum_{uv \in E(G)} c(uv). The smallest integer kk such that GG has a proper total kk-coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} χΣ(G)\chi''_{\Sigma}(G). Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that χΣ(G)Δ(G)+3\chi''_{\Sigma}(G)\leq \Delta(G)+3 for any simple graph with maximum degree Δ(G)\Delta(G). We prove this bound to be asymptotically correct by showing that χΣ(G)Δ(G)(1+o(1))\chi''_{\Sigma}(G)\leq \Delta(G)(1+o(1)). The main idea of our argument relies on Przyby{\l}o's proof (2014) for neighbor sum distinguishing edge-coloring
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