Online version of the theorem of Thue

A sequence S is nonrepetitive if no two adjacent blocks of S are the same. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3 symbols. We consider the online variant of this result in which a nonrepetitive sequence is constructed during a play between two players: Bob is choosing a position in a sequence and Alice is inserting a symbol on that position taken from a fixed set A. The goal of Bob is to force Alice to create a repetition, and if he succeeds, then the game stops. The goal of Alice is naturally to avoid that and thereby to construct a nonrepetitive sequence of any given length. We prove that Alice has a strategy to play arbitrarily long provided the size of the set A is at least 12. This is the online version of the Theorem of Thue. The proof is based on nonrepetitive colorings of outerplanar graphs. On the other hand, one can prove that even over 4 symbols Alice has no chance to play for too long. The minimum size of the set of symbols needed for the online version of Thue's theorem remains unknown

Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196

Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult

We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem

The Incidence Chromatic Number of Toroidal Grids

An incidence in a graph $G$ is a pair $(v,e)$ with $v \in V(G)$ and $e \in E(G)$, such that $v$ and $e$ are incident. Two incidences $(v,e)$ and $(w,f)$ are adjacent if $v=w$, or $e=f$, or the edge $vw$ equals $e$ or $f$. The incidence chromatic number of $G$ is the smallest $k$ for which there exists a mapping from the set of incidences of $G$ to a set of $k$ colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid $T_{m,n}=C_m\Box C_n$ equals 5 when $m,n \equiv 0 \pmod 5$ and 6 otherwise.Comment: 16 page

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

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

Planar graph coloring avoiding monochromatic subgraphs : trees and paths make it difficult

We consider the problem of coloring a planar graph with the minimum number of colors so that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem. We present a complete picture for the case with a single forbidden connected (induced or noninduced) subgraph. The 2-coloring problem is NP-hard if the forbidden subgraph is a tree with at least two edges, and it is polynomially solvable in all other cases. The 3-coloring problem is NP-hard if the forbidden subgraph is a path with at least one edge, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles. In particular, we prove that it is NP-complete to decide if a planar graph can be 2-colored so that no cycle of length at most 5 is monochromatic