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APC: Only $99 Submit 2020 ( 6 ) 2019 ( 371 ) 2018 ( 516 ) 2017 ( 472 ) Search Results: 1 - 10 of 379721 matches for " János Barát " All listed articles are free for downloading (OA Articles)  Page 1 /379721 Display every page 5 10 20 Item  Mathematics , 2011, DOI: 10.1002/jgt.21695 Abstract: A sequence$s_1,s_2,...,s_k,s_1,s_2,...,s_k$is a repetition. A sequence$S$is nonrepetitive, if no subsequence of consecutive terms of$S$form a repetition. Let$G$be a vertex colored graph. A path of$G$is nonrepetitive, if the sequence of colors on its vertices is nonrepetitive. If$G$is a plane graph, then a facial nonrepetitive vertex coloring of$G$is a vertex coloring such that any facial path is nonrepetitive. Let$\pi_f(G)$denote the minimum number of colors of a facial nonrepetitive vertex coloring of$G$. Jendro\vl and Harant posed a conjecture that$\pi_f(G)$can be bounded from above by a constant. We prove that$\pi_f(G)\le 24$for any plane graph$G$.  Mathematics , 2005, Abstract: A vertex colouring of a graph is \emph{nonrepetitive on paths} if there is no path$v_1,v_2,...,v_{2t}$such that v_i and v_{t+i} receive the same colour for all i=1,2,...,t. We determine the maximum density of a graph that admits a k-colouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4-colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree$\Delta$has a$f(\Delta)$-colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree$\Delta$has a$O(k\Delta)$-colouring that is nonrepetitive on paths, and a$O(k\Delta^3)$-colouring that is nonrepetitive on walks.  Mathematics , 2009, Abstract: We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on$n$vertices, and Avoider's goal is to keep his graph outerplanar, diamond-free and$k$-degenerate, respectively. It is clear that all three games are Enforcer's wins, and our main interest lies in determining the largest number of moves Avoider can play before losing. Extremal graph theory offers a general upper bound for the number of Avoider's moves. As it turns out, for all three games we manage to obtain a lower bound that is just an additive constant away from that upper bound. In particular, we exhibit a strategy for Avoider to keep his graph outerplanar for at least$2n-8$moves, being just 6 short of the maximum possible. A diamond-free graph can have at most$d(n)=\lceil\frac{3n-5}{2}\rceil$edges, and we prove that Avoider can play for at least$d(n)-3$moves. Finally, if$k$is small compared to$n$, we show that Avoider can keep his graph$k$-degenerate for as many as$e(n)$moves, where$e(n)$is the maximum number of edges a$k$-degenerate graph can have.  Mathematics , 2012, Abstract: We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Bar\'at and Thomassen: for each tree$T$, there exists a natural number$k_T$such that if$G$is a$k_T$-edge-connected graph, and$|E(T)|$divides$|E(G)|$, then$E(G)$has a decomposition into copies of$T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. Let$Y$be the unique tree with degree sequence$(1,1,1,2,3)$. We prove that if$G$is a 191-edge-connected graph of size divisible by 4, then$G$has a$Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star.  Mathematics , 2015, Abstract: A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. Brandenburg et al. showed that there are maximal 1-planar graphs with only$\frac{45}{17}n + O(1)\approx 2.647n$edges and maximal 1-plane graphs with only$\frac{7}{3}n+O(1)\approx 2.33n$edges. On the other hand, they showed that a maximal 1-planar graph has at least$\frac{28}{13}n-O(1)\approx 2.15n-O(1)$edges, and a maximal 1-plane graph has at least$2.1n-O(1)$edges. We improve both lower bounds to$\frac{20n}{9}\approx 2.22n$.  Mathematics , 2009, Abstract: Albertson conjectured that if a graph$G$has chromatic number$r$then its crossing number is at least as much as the crossing number of$K_r$. Albertson, Cranston, and Fox verified the conjecture for$r\le 12$. We prove the statement for$r\le 16$.  Mathematics , 2014, DOI: 10.1007/s00373-015-1575-9 Abstract: A famous conjecture of Ryser is that in an$r$-partite hypergraph the covering number is at most$r-1$times the matching number. If true, this is known to be sharp for$r$for which there exists a projective plane of order$r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely$r=7$. For$r\in\{6,7\}$, we find the minimal number$f(r)$of edges in an intersecting$r$-partite hypergraph that has covering number at least$r-1$. We find that$f(r)$is achieved only by linear hypergraphs for$r\le5$, but that this is not the case for$r\in\{6,7\}$. We also improve the general lower bound on$f(r)$, showing that$f(r)\ge 3.052r+O(1)$. We show that a stronger form of Ryser's conjecture that was used to prove the$r=3$case fails for all$r>3$. We also prove a fractional version of the following stronger form of Ryser's conjecture: in an$r$-partite hypergraph there exists a set$S$of size at most$r-1$, contained either in one side of the hypergraph or in an edge, whose removal reduces the matching number by 1.  Computer Science , 2011, Abstract: The List Hadwiger Conjecture asserts that every$K_t$-minor-free graph is$t$-choosable. We disprove this conjecture by constructing a$K_{3t+2}$-minor-free graph that is not$4t$-choosable for every integer$t\geq 1$.  Mathematics , 2010, Abstract: For a property$\Gamma$and a family of sets$\cF$, let$f(\cF,\Gamma)$be the size of the largest subfamily of$\cF$having property$\Gamma$. For a positive integer$m$, let$f(m,\Gamma)$be the minimum of$f(\cF,\Gamma)$over all families of size$m$. A family$\cF$is said to be$B_d$-free if it has no subfamily$\cF'=\{F_I: I \subseteq [d]\}$of$2^d$distinct sets such that for every$I,J \subseteq [d]$, both$F_I \cup F_J=F_{I \cup J}$and$F_I \cap F_J = F_{I \cap J}$hold. A family$\cF$is$a$-union free if$F_1\cup ... F_a \neq F_{a+1}$whenever$F_1,..,F_{a+1}$are distinct sets in$\FF$. We verify a conjecture of Erd\H os and Shelah that$f(m, B_2\text{\rm -free})=\Theta(m^{2/3})$. We also obtain lower and upper bounds for$f(m, B_d\text{\rm -free})$and$f(m,a\text{\rm -union free})\$.
 Mathematics , 2009, Abstract: Nancy G. Kinnersley and Michael A. Langston has determined the excluded minors for the class of graphs with path-width at most two by computer. Their list consisted of 110 graphs. Such a long list is difficult to handle and gives no insight to structural properties. We take a different route, and concentrate on the building blocks and how they are glued together. In this way, we get a characterization of 2-connected and 2-edge-connected graphs with path-width at most two. Along similar lines, we sketch the complete characterization of graphs with path-width at most two.
 Page 1 /379721 Display every page 5 10 20 Item