Graph multicoloring reduction methods and application to McDiarmid-Reed's Conjecture

A $(a,b)$-coloring of a graph $G$ associates to each vertex a set of $b$ colors from a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoints. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le a/b\le 3$. In particular, we prove necessary and sufficient conditions for the existence of a $(a,b)$-coloring of a path with prescribed color-sets on its end-vertices. Other more complex $(a,b)$-colorability reductions are presented. The utility of these tools is exemplified on finite triangle-free induced subgraphs of the triangular lattice. Computations on millions of such graphs generated randomly show that our tools allow to find (in linear time) a $(9,4)$-coloring for each of them. Although there remain few graphs for which our tools are not sufficient for finding a $(9,4)$-coloring, we believe that pursuing our method can lead to a solution of the conjecture of McDiarmid-Reed.Comment: 27 page

Az új algoritmusok és kódolási eljárások alkalmazása a mobil hírközlésben és informatikában = Application of new algorithms and coding procedures in mobile communications and computing

A kutatási munka során az alábbi résztémákban értünk el eredményeket: - mobil IP, - all IP hálózatok, - útkeresési algoritmusok, - hívásátadási algoritmusok, - mobil technológiák együttműködése, - a szolgáltatás minősége (QoS), - a mobil és informatikai hálózatok és rendszerek biztonsági kérdései, - több-felhasználós vétel, - kódosztásos többszörös hozzáférés, - forgalmi modellezés, - kódkonstrukció kódosztásos technológiákhoz, - kvantum számítástechnikai eljárások, - gráfelmélet, - kombinatorikus optimalizálás. A fenti szakterületeken végzett kutatásaink eredményei közül azokat emeljük ki, amelyeket az alábbi témákban értünk el: - A heterogén mobil hálózatok együttműködési problémái, - A mobil Internet Protokoll alkalmazásával kapcsolatos vizsgálatok, - Többfelhasználós detekciós módszerek a kódosztásos többszörös hozzáféréses mobil rendszerekben, - A heterogén mobil hálózatok forgalmi modellezése, - A mobil informatikai és távközlési hálózatok, rendszerek és szolgáltatások - biztonsági kérdései, - Kvantum számítástechnika és mérnöki alkalmazásai, - Útkeresési és csatornakijelölési algoritmusok fejlesztése és vizsgálata mobil hálózatok számára, alkalmazott gráfelmélet. A kutatásban résztvevők az eredményeket három megvédett PhD disszertációban, egy benyújtás előtt álló akadémiai doktori értekezésben és több beadás előtt álló PhD értekezésben használták fel. A tudományos iskola publikációs listája 135 elemből áll. | The members of the Scientific School have got new results in the following scientific fields: - Mobile IP, all IP networks, - Routing algorithms, - Hand-over algorithms, - Interworking of heterogeneous mobile technologies, - Quality of services (QoS), - Security problems of mobile and information networks and systems, - Multi-user detection, - Code division multiple access, - Traffic modeling, - Code construction for code division technologies, - Quantum computing, - Graph theory, - Combinatorial optimization. On the above mentioned scientific field we have the most important results in the following areas: - Interoperability issues of heterogeneous mobile networks, - Investigations on the applicability of mobile Internet Protocol, - Multi-user detection methods in code division multiple access systems, - Traffic models of heterogeneous mobile networks, - Security issues of mobile information and telecommunication networks, systems and services, - Quantum computing and its engineering applications, - Development and research of routing and channel assigning algorithms for mobile networks, application of the graph theory. The participants of the research used their results in three defended PhD theses, in a dissertation for DSc title, and in some other PhD theses before the final process. The number of the publications of the Scientific School is 135

On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

Online Multi-Coloring with Advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

Finding Large H-Colorable Subgraphs in Hereditary Graph Classes

“First Published in SIAM Journal on Discrete Mathematics in 35, 4, 2021, published by the Society for Industrial and Applied Mathematics (SIAM)” and the copyright notice as stated in the article itself (e.g., “Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.”')We study the Max Partial H-Coloring problem: given a graph G, find the largest induced subgraph of G that admits a homomorphism into H, where H is a fixed pattern graph without loops. Note that when H is a complete graph on k vertices, the problem reduces to finding the largest induced k-colorable subgraph, which for k=2 is equivalent (by complementation) to Odd Cycle Transversal. We prove that for every fixed pattern graph H without loops, Max Partial H-Coloring can be solved in {P5,F}-free graphs in polynomial time, whenever F is a threshold graph; in {P5,bull}-free graphs in polynomial time; in P5-free graphs in time nO(ω(G)); and in {P6,1−subdividedclaw}-free graphs in time nO(ω(G)3). Here, n is the number of vertices of the input graph G and ω(G) is the maximum size of a clique in G. Furthermore, by combining the mentioned algorithms for P5-free and for {P6,1−subdividedclaw}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for Max Partial H-Coloring in these classes of graphs. Finally, we show that even a restricted variant of Max Partial H-Coloring is NP-hard in the considered subclasses of P5-free graphs if we allow loops on H.The first author’s material is based upon work supported in part by the U.S. ArmyResearch Office under grant W911NF-16-1-0404 and by NSF grant DMS-1763817. The third author’swork is a part of project TOTAL that has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (grant 677651).The fourth author was supported by Polish National Science Centre grant 2018/31/D/ST6/00062.The fifth author’s material is based upon work supported by the National Science Foundation underaward DMS-1802201

Improved distributed algorithms for coloring interval graphs with application to multicoloring trees

Post-print (lokagerð höfundar)We give a distributed (1+eps)-approximation algorithm for the minimum vertex coloring problem on interval graphs, which runs in the LOCAL model and operates in O((1/eps) log* n) rounds. If nodes are aware of their interval representations, then the algorithm can be adapted to the CONGEST model using the same number of rounds. Prior to this work, only constant factor approximations using O(log* n) rounds were known. Linial's ring coloring lower bound implies that the dependency on log* n cannot be improved. We further prove that the dependency on 1/eps is also optimal. To obtain our CONGEST model algorithm, we develop a color rotation technique that may be of independent interest. We demonstrate that color rotations can also be applied to obtain a (1+eps)-approximate multicoloring of directed trees in O((1/eps)log* n) rounds.Magnus M. Halldorsson is supported by grants 152679-05 and 174484-05 from the Icelandic Research Fund. Christian Konrad is supported by the Centre for Discrete Mathematics and its Applications (DIMAP) at Warwick University and by EPSRC award EP/N011163/1."Peer Reviewed