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Entropies, Capacities, and Colorings of Graphs

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CORE Metadata, citation and similar papers at core.ac.uk Provided by REAL-d Entropies, Capacities, and Colorings of Graphs Simonyi G´abor MTA R´enyi Alfr´ed Matematikai Kutat´oint´ezet H-1364 Budapest, Pf. 127. [email protected] Budapest, 2006 Contents Bevezet´es ´es ´attekint´es (Summary in Hungarian) iii 1 Graph Entropies 1 1.1 Entropysplittinghypergraphs . .... 1 1.1.1 Introduction............................... 1 1.1.2 Basicdefinitions............................. 2 1.1.3 Splitting3-uniformhypergraphs . ... 3 1.1.4 The case k 4 ............................. 9 1.1.5 Connectionswithcographs.≥ . 11 1.2 Imperfectionratioandgraphentropy . ..... 13 1.2.1 Somepreliminaries . .. .. 13 1.2.2 An entropy formula for the imperfection ratio . ..... 15 1.2.3 Dilation ratio and entropy of convex corners . ..... 18 1.3 Witsenhausen rate of multiple sources . ...... 22 1.3.1 Introduction............................... 22 1.3.2 Thegraphtheorymodel . 22 1.3.3 Probabilisticgraphinvariants . ... 24 1.3.4 ProofofTheorem1.3.1. 27 2 Graph Capacities 29 2.1 Differentcapacitiesofadigraph . .... 29 2.1.1 Introduction............................... 29 2.1.2 Waterfalls ................................ 33 2.1.3 Independencegraphs . 34 2.1.4 Waterfallsinundirectedgraphs . .. 37 2.1.5 Whenaretheboundstight? . 42 2.2 Orientations of self-complementary graphs and the relation of Sperner and Shannoncapacities ............................... 47 2.2.1 Introduction............................... 47 2.2.2 D(G) versus C(G) ........................... 48 2.2.3 Self-complementarygraphs. 49 2.2.4 Consequences for Sperner capacity . .. 52 i 2.2.5 Furtherremarks............................. 53 2.3 Local chromatic number and Sperner capacity . ...... 55 2.3.1 Introduction............................... 55 2.3.2 Local chromatic number for directed graphs . .... 56 2.3.3 Spernercapacity ............................ 56 2.3.4 Mainresult ............................... 58 2.3.5 Application:oddcycles. 59 2.3.6 Theundirectedcase .......................... 61 2.3.7 Fractionalcolorings. 62 2.3.8 Fractionalcovers ............................ 67 3 Graph Colorings 71 3.1 Local chromatic number, Ky Fan’s theorem, and circular colorings . 71 3.1.1 Introduction............................... 71 3.1.2 Results.................................. 73 3.1.3 Lowerbound .............................. 76 3.1.4 Upperbound .............................. 81 3.1.5 Circularcolorings............................ 95 3.1.6 Furtherremarks. .. .. .100 3.2 Colorful subgraphs in Kneser-like graphs . .......103 3.2.1 Introduction...............................103 3.2.2 A generalization of Dol’nikov’s theorem . .104 3.2.3 Applying a theorem of Tucker and Bacon . 106 Bibliography 115 ii Bevezet´es ´es ´attekint´es (Summary in Hungarian) Az egyik legfontosabb ´es legt¨obbet vizsg´alt gr´afelm´eleti param´eter a kromatikus sz´am, azon sz´ınek minim´alis sz´ama, melyekkel a gr´af cs´ucsai kisz´ınezhet˝ok ´ugy, hogy szomsz´edos cs´ucsok sz´ıne k¨ul¨onb¨oz˝olegyen. A G gr´af χ(G) kromatikus sz´am´anak egyik legterm´eszetesebb als´obecsl´ese a gr´af ω(G) klikksz´ama, a cs´ucshalmaz legnagyobb olyan r´eszhalmaz´anak elemsz´ama, amelyben min- den cs´ucsp´ar ¨ossze van k¨otve. Ez a becsl´es sokszor nagyon gyenge, egy h´aromsz¨ogmentes gr´af kromatikus sz´ama is lehet tetsz˝olegesen nagy, ezt bizony´ıtja pl. Mycielski [123] konstrukci´oja. Az ´ertekez´esben fontos szerepet j´atszik gr´afok konorm´alisnak nevezett szorzata, illetve hatv´anya, melynek bevezet´es´et egyebek mellett sz´amos inform´aci´oelm´eleti k´erd´es mo- tiv´alja. E m˝uvelet sor´an a klikksz´am ´es a kromatikus sz´am ellent´etesen viselkedik: el˝obbi szupermultiplikat´ıv, ut´obbi szubmultiplikat´ıv erre a szorz´asra n´ezve, melynek defin´ıci´oja a k¨ovetkez˝o. Defin´ıci´o. Az F ´es G gr´afok konorm´alis szorzata az az F G gr´af, melyre · V (F G) = V (F ) V (G) · × E(F G) = (f,g), (f ′,g′) : f, f ′ E(F ) vagy g,g′ E(G) . · {{ } { } ∈ { } ∈ } Gt a G gr´af ¨onmag´aval vett t-szeres konorm´alis szorzata, amit G t-edik konorm´alis hatv´any´anak h´ıvunk. A most defini´alt hatv´anyoz´asra ´ugy ´erdemes gondolni, hogy amennyiben G ´elei a cs´ucsok valamif´ele megk¨ul¨onb¨oztethet˝os´eg´et jelentik, akkor a konorm´alis hatv´anyoz´as ezt a rel´aci´ot terjeszti ki a cs´ucsok t hossz´usorozataira: k´et ilyen sorozat pontosan akkor megk¨ul¨onb¨oztethet˝o, ha legal´abb egy koordin´at´aban az. Nem neh´ez bel´atni, hogy, mint fent eml´ıtett¨uk, tetsz˝oleges F ´es G gr´afra fenn´all az ω(F G) ω(F )ω(G) valamint a χ(F G) χ(F )χ(G) · ≥ · ≤ egyenl˝otlens´eg. A fentib˝ol k¨ovetkezik, hogy a χ∗(G) := lim t χ(Gt) t→∞ p iii ´es a c(G) := lim t ω(Gt) t→∞ hat´ar´ert´ekek egyar´ant l´eteznek, ´es r´ajuk ω(G) pc(G) χ∗(G) χ(G) teljes¨ul. ≤ ≤ ≤ Az els˝ok´ent fel´ırt χ∗(G) hat´ar´ert´ek j´ol ismert mennyis´eg. A kromatikus sz´am fel´ırhat´o egy eg´esz´ert´ek˝uprogramoz´asi feladat megold´asak´ent. Ennek val´os relax´aci´oj´at megoldva jutunk a frakcion´alis kromatikus sz´am fogalm´ahoz, mely McEliece ´es Posner [120] egy t´etel´eb˝ol ad´od´oan (ld. Berge ´es Simonovits [19] dolgozat´at is) megegyezik a fenti χ∗(G) hat´ar´ert´ekkel. A klikksz´am is fel´ırhat´oeg´esz´ert´ek˝uprogramoz´asi feladat megold´asak´ent, ennek val´os relax´aci´oja a frakcion´alis klikksz´am fogalm´ahoz vezet, amelynek ´ert´eke a line´aris prog- ramoz´as dualit´ast´etele r´ev´en mindig azonos a frakcion´alis kromatikus sz´am, teh´at χ∗(G) ´ert´ek´evel. V´arhat´ovolna mindezek alapj´an, hogy c(G) is ezzel a k¨oz¨os ´ert´ekkel legyen egyenl˝o. Ez azonban nincs ´ıgy. A c(G) mennyis´eg, pontosabban annak logaritmusa, Shannon [139] inform´aci´oelm´eleti vizsg´alataiban bukkant fel el˝osz¨or. Defin´ıci´o. (Shannon [139]) Egy G gr´af (logaritmikus) Shannon kapacit´asa1 a 1 C(G) = lim log ω(Gt) t 2 mennyis´eg. A logaritm´al´as oka az inform´aci´oelm´eleti h´att´er, a fenti mennyis´eg ugyanis egy zajos csatorna bitekben m´ert ´un. z´er´o-hiba kapacit´as´at fejezi ki. Innen ad´odik az is, hogy a logaritmus alapja 2. A tov´abbiakban is minden logaritmus kettes alap´ulesz, ezent´ul ezt nem ´ırjuk ki. A Shannon kapacit´as a modern kombinatorika egyik k¨ul¨on¨osen ´erdekes fogalma. Vizsg´alat´at sz´amos v´aratlan kapcsolat, valamint n´emely vele kapcsolatos probl´ema meglep˝oneh´ezs´ege egyar´ant indokolja. Az a tal´an ´artatlannak l´atsz´ok´erd´es p´eld´aul, hogy egy h´aromsz¨ogmentes gr´afra a C(G) ´ert´ek lehet-e tetsz˝olegesen nagy, ekvivalens Erd˝osnek egy m´aig megoldatlan probl´em´aj´aval, mely azt k´erdezi, hogy az R(3 : t) := R(3, 3,..., 3) Ramsey sz´am (az a legkisebb r sz´am, amire a Kr teljes gr´af ´eleit t sz´ınnel sz´ınezve biztosan keletkezik egysz´ın˝uh´aromsz¨og) gyorsabban n˝o-e, mint b´armilyen r¨ogz´ıtett c konstans t- edik hatv´anya (ld. Erd˝os, McEliece ´es Taylor [49], Alon ´es Orlitsky [6], valamint Rosenfeld ´es Neˇsetˇril [125] cikkeit). Shannon [139] meghat´arozta minden legfeljebb 4 cs´ucs´ugr´af Shannon kapacit´as´at ´es az 5 cs´ucs´uak´et is egy kiv´etellel. Az 5 hossz´us´ag´u C5 k¨or Shannon kapacit´as´ar´ol csak 23 ´evvel k´es˝obb bizony´ıtotta be Lov´asz [109], hogy a Shannon ´altal megadott als´okorl´attal, log √5- tel egyenl˝o. A Shannon kapacit´as probl´ema neh´ezs´eg´et a fentieken t´ul az is j´ol mutatja, 1Megjegyezz¨uk, hogy sz´amos t´argyal´as a C(G¯) mennyis´eget nevezi a G gr´af Shannon kapacit´as´anak, ahol G¯ a G gr´af komplementere. Maga Shannon is ezt a nyelvezetet haszn´alja, mi az´ert nem ezt k¨ovetj¨uk, mert az ir´any´ıtott gr´afok kapacit´asainak t´argyal´asa´ıgy term´eszetesebb lesz. Err˝ol b˝ovebben ld. a dolgozat 1.5. Megjegyz´es´et. iv hogy az ¨otn´el hosszabb p´aratlan k¨or¨okre mindm´aig ismeretlen a C(G) ´ert´ek, ´es m´eg azt is csak 2003-ban igazolta Bohman ´es Holzman [24], hogy C(C2k+1) > log 2 minden k-ra, azaz minden h´aromn´al hosszabb p´aratlan k¨or Shannon kapacit´asa meghaladja a klikksz´am´ab´ol ad´od´oegyszer˝uals´okorl´atot. A Shannon kapacit´as vizsg´alata ´altal inspir´alva vezette be Berge [15, 16, 17] a perfekt gr´afokat (ld. Ram´ırez-Alfons´ın ´es Berge [18]). Defin´ıci´o. (Berge [16]) Egy G gr´af perfekt, ha minden G′ fesz´ıtett r´eszgr´afj´ara χ(G′)= ω(G′) teljes¨ul. A perfekt gr´afok rendk´ıv¨ul fontos ´es sokat vizsg´alt gr´afoszt´alyt alkotnak. Ennek legf˝obb oka az, hogy kapcsolatot teremtenek olyan l´atsz´olag t´avolabbi ter¨uletek k¨oz¨ott, mint a gr´afelm´elet, a poli´ederes kombinatorika ´es az inform´aci´oelm´elet. Sz´amos ilyen kapcsolatot r´eszletesen t´argyal a Ram´ırez-Alfons´ın ´es Reed ´altal szerkesztett [128] k¨onyv, valamint Schrijver [136] monument´alis monogr´afi´aj´anak h´arom perfekt gr´afokr´ol sz´ol´ofe- jezete. Igen sok ´erdekes gr´af perfekt. Ilyenek p´eld´aul a p´aros gr´afok ´es ´elgr´afjaik, az intervallumgr´afok, vagy a r´eszben rendezett halmazokhoz rendelhet˝o´un.
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