A new approach to nonrepetitive sequences

A sequence is nonrepetitive if it does not contain two adjacent identical blocks. The remarkable construction of Thue asserts that 3 symbols are enough to build an arbitrarily long nonrepetitive sequence. It is still not settled whether the following extension holds: for every sequence of 3-element sets $L_1,..., L_n$ there exists a nonrepetitive sequence $s_1, ..., s_n$ with $s_i\in L_i$. Applying the probabilistic method one can prove that this is true for sufficiently large sets $L_i$. We present an elementary proof that sets of size 4 suffice (confirming the best known bound). The argument is a simple counting with Catalan numbers involved. Our approach is inspired by a new algorithmic proof of the Lov\'{a}sz Local Lemma due to Moser and Tardos and its interpretations by Fortnow and Tao. The presented method has further applications to nonrepetitive games and nonrepetitive colorings of graphs.Comment: 5 pages, no figures.arXiv admin note: substantial text overlap with arXiv:1103.381

Gráfszínezések és gráfok felbontásai = Colorings and decompositions of graphs

A nem-ismétlő színezéseket a véletlen módszer alkalmazhatósága miatt kezdték el vizsgálni. Felső korlátot adtunk a színek számára, amely a maximum fok és a favastagság lineáris függvénye. Olyan színezéseket is vizsgáltunk, amelyek egy síkgráf oldalain nem-ismétlők. Sejtés volt, hogy véges sok szín elég. Ezt bizonyítottuk 24 színnel. A kromatikus számot és a metszési számot algoritmikusan nehéz meghatározni. Ezért meglepő Albertson egy friss sejtése, amely kapcsolatot állít fel közöttük: ha egy gráf kromatikus száma r, akkor metszési száma legalább annyi, mint a teljes r csúcsú gráfé. Bizonyítottuk a sejtést, ha r<3.57n, valamint ha 12<r<17. Ez utóbbi azért érdekes, mert a teljes r csúcsú gráf metszési száma csak r<13 esetén ismert. A témakör legfontosabb nyitott kérdése a Hadwiger-sejtés, mely szerint minden r-kromatikus gráf tartalmazza a teljes r csúcsú gráfot minorként. Ennek általánosításaként fogalmazták meg a lista színezési Hadwiger sejtést: ha egy gráf nem tartalmaz teljes r csúcsú gráfot minorként, akkor az r-lista színezhető. Megmutattuk, hogy ez a sejtés hamis. Legalább cr színre szükségünk van bizonyos gráfokra, ahol c=4/3. Thomassennel vetettük fel azt a kérdést, hogy milyen feltétel garantálja, hogy G élei felbonthatók egy adott T fa példányaira. Legyen Y az a fa, melynek fokszámsorozata (1,1,1,2,3). Megmutattuk, hogy minden 287-szeresen élösszefüggő fa felbomlik Y-okra, ha az élszám osztható 4-gyel. | Nonrepetitive colorings often use the probabilistic method. We gave an upper bound as a linear function of the maximum degree and the tree-width. We also investigated colorings, which are nonrepetitive on faces of plane graphs. As conjectured, a finite number of colors suffice. We proved it by 24 colors. The chromatic and crossing numbers are both difficult to compute. The recent Albertson's conjecture is a surprising relation between the two: if the chromatic number is r, then the crossing number is at least the crossing number of the complete graph on r vertices. We proved this claim, if r<3.57n, or 12<r<17. The latter is remarkable, since the crossing number of the complete graph is only known for r<13. The most important open question of the field is Hadwiger's conjecture: every r-chromatic graph contains as a minor the complete graph on r vertices. As a generalisation, the following is the list coloring Hadwiger conjecture: if a graph does not contain as a minor the complete graph on r vertices , then the graph is r-list colorable. We proved the falsity of this claim. In our examples, at least cr colors are necessary, where c=4/3. Decomposition of graphs is well-studied. Thomassen and I posed the question of a sufficient connectivity condition, which guaranties a T-decomposition. Let Y be the tree with degree sequence (1,1,1,2,3). We proved every 287-edge connected graph has a Y-decomposition, if the size is divisible by four

Extensions and reductions of square-free words

A word is square-free if it does not contain a nonempty word of the form $XX$ as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long square-free words over a $3$-letter alphabet. We study square-free words with additional properties involving single-letter deletions and extensions of words. A square-free word is steady if it remains square-free after deletion of any single letter. We prove that there exist infinitely many steady words over a $4$-letter alphabet. We also demonstrate that one may construct steady words of any length by picking letters from arbitrary alphabets of size $7$ assigned to the positions of the constructed word. We conjecture that both bounds can be lowered to $4$, which is best possible. In the opposite direction, we consider square-free words that remain square-free after insertion of a single (suitably chosen) letter at every possible position in the word. We call them bifurcate. We prove a somewhat surprising fact, that over a fixed alphabet with at least three letters, every steady word is bifurcate. We also consider families of bifurcate words possessing a natural tree structure. In particular, we prove that there exists an infinite tree of doubly infinite bifurcate words over alphabet of size $12$.Comment: 11 pages, 1 figur

The Lefthanded Local Lemma characterizes chordal dependency graphs

Shearer gave a general theorem characterizing the family \LLL of dependency graphs labeled with probabilities $p_v$ which have the property that for any family of events with a dependency graph from \LLL (whose vertex-labels are upper bounds on the probabilities of the events), there is a positive probability that none of the events from the family occur. We show that, unlike the standard Lov\'asz Local Lemma---which is less powerful than Shearer's condition on every nonempty graph---a recently proved `Lefthanded' version of the Local Lemma is equivalent to Shearer's condition for all chordal graphs. This also leads to a simple and efficient algorithm to check whether a given labeled chordal graph is in \LLL.Comment: 12 pages, 1 figur

Bad News for Chordal Partitions

Reed and Seymour [1998] asked whether every graph has a partition into induced connected non-empty bipartite subgraphs such that the quotient graph is chordal. If true, this would have significant ramifications for Hadwiger's Conjecture. We prove that the answer is `no'. In fact, we show that the answer is still `no' for several relaxations of the question