734 research outputs found
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
Local colourings and monochromatic partitions in complete bipartite graphs
We show that for any -local colouring of the edges of the balanced
complete bipartite graph , its vertices can be covered with at
most~ disjoint monochromatic paths. And, we can cover almost all vertices of
any complete or balanced complete bipartite -locally coloured graph with
disjoint monochromatic cycles.\\ We also determine the -local
bipartite Ramsey number of a path almost exactly: Every -local colouring of
the edges of contains a monochromatic path on vertices.Comment: 18 page
Minimum degree conditions for monochromatic cycle partitioning
A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
-edge-coloured complete graph has a partition into
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant such that
any -edge-coloured graph on vertices with minimum degree at least has a partition into monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix
Partitioning random graphs into monochromatic components
Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every -colored
complete graph can be partitioned into at most monochromatic components;
this is a strengthening of a conjecture of Lov\'asz (1975) in which the
components are only required to form a cover. An important partial result of
Haxell and Kohayakawa (1995) shows that a partition into monochromatic
components is possible for sufficiently large -colored complete graphs.
We start by extending Haxell and Kohayakawa's result to graphs with large
minimum degree, then we provide some partial analogs of their result for random
graphs. In particular, we show that if , then a.a.s. in every -coloring of there exists
a partition into two monochromatic components, and for if , then a.a.s. there exists an -coloring
of such that there does not exist a cover with a bounded number of
components. Finally, we consider a random graph version of a classic result of
Gy\'arf\'as (1977) about large monochromatic components in -colored complete
graphs. We show that if , then a.a.s. in every
-coloring of there exists a monochromatic component of order at
least .Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics
Volume 24, Issue 1 (2017) Paper #P1.1
Vertex covering with monochromatic pieces of few colours
In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every -colouring of the
edges of , there is a vertex cover by monochromatic paths of
the same colour, which is optimal up to a constant factor. The main goal of
this paper is to study the natural multi-colour generalization of this problem:
given two positive integers , what is the smallest number
such that in every colouring of the edges of with
colours, there exists a vertex cover of by
monochromatic paths using altogether at most different colours? For fixed
integers and as , we prove that , where is the chromatic number of
the Kneser gr aph . More generally, if one replaces by
an arbitrary -vertex graph with fixed independence number , then we
have , where this time around is the
chromatic number of the Kneser hypergraph . This
result is tight in the sense that there exist graphs with independence number
for which . This is in sharp
contrast to the case , where it follows from a result of S\'ark\"ozy
(2012) that depends only on and , but not on
the number of vertices. We obtain similar results for the situation where
instead of using paths, one wants to cover a graph with bounded independence
number by monochromatic cycles, or a complete graph by monochromatic
-regular graphs
Extremal problems on special graph colorings
In this thesis, we study several extremal problems on graph colorings. In particular, we study monochromatic connected matchings, paths, and cycles in 2-edge colored graphs, packing colorings of subcubic graphs, and directed intersection number of digraphs.
In Chapter 2, we consider monochromatic structures in 2-edge colored graphs. A matching M in a graph G is connected if all the edges of M are in the same component of G. Following Łuczak, there are a number of results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in 2-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings, which is used to find necessary and sufficient conditions on the existence of monochromatic paths and cycles: for every fixed s and large n, we describe all values of n_1, ...,n_s such that for every 2-edge-coloring of the complete s-partite graph K_{n_1, ...,n_s} there exists a monochromatic (i) cycle C_{2n} with 2n vertices, (ii) cycle C_{at least 2n} with at least 2n vertices, (iii) path P_{2n} with 2n vertices, and (iv) path P_{2n+1} with 2n+1 vertices. Our results also imply for large n of the conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi that for every 2-edge-coloring of the complete 3-partite graph K_{n,n,n} there is a monochromatic path P_{2n+1}.
Moreover, we prove that for every sufficiently large n, if n = 3t+r where r in {0,1,2} and G is an n-vertex graph with minimum degree at least (3n-1)/4, then for every 2-edge-coloring of G, either there are cycles of every length {3, 4, 5, ..., 2t+r} of the same color, or there are cycles of every even length {4, 6, 8, ..., 2t+2} of the same color. This result is tight and implies the conjecture of Schelp that for every sufficiently large n, every (3n-1)-vertex graph G with minimum degree larger than 3|V(G)|/4, in each 2-edge-coloring of G there exists a monochromatic path P_{2n} with 2n vertices. It also implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for every positive integer n of the form n=3t+r where r in {0,1,2} and every n-vertex graph G with minimum degree at least 3n/4, in each 2-edge-coloring of G there exists a monochromatic cycle of length at least 2t+r.
In Chapter 3, we consider a collection of special vertex colorings called packing colorings. For a sequence of non-decreasing positive integers S = (s_1, ..., s_k), a packing S-coloring is a partition of V(G) into sets V_1, ..., V_k such that for each integer i in {1, ..., k} the distance between any two distinct x,y in V_i is at least s_i+1. The smallest k such that G has a packing (1,2, ..., k)-coloring is called the packing chromatic number of G and is denoted by \chi_p(G). The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We show that for every fixed k and g at least 2k+2, almost every n-vertex cubic graph of girth at least g has the packing chromatic number greater than k, which answers the previous question in the negative. Moreover, we work towards the conjecture of Brešar, Klavžar, Rall and Wash that the packing chromatic number of 1-subdivision of subcubic graphs are bounded above by 5. In particular, we show that every subcubic graph is (1,1,2,2,3,3,k)-colorable for every integer k at least 4 via a coloring in which color k is used at most once, every 2-degenerate subcubic graph is (1,1,2,2,3,3)-colorable, and every subcubic graph with maximum average degree less than 30/11 is packing (1,1,2,2)-colorable.
Furthermore, while proving the packing chromatic number of subcubic graphs is unbounded, we also consider improving upper bound on the independence ratio, alpha(G)/n, of cubic n-vertex graphs of large girth. We show that ``almost all" cubic labeled graphs of girth at least 16 have independence ratio at most 0.454.
In Chapter 4, we introduce and study the directed intersection representation of digraphs. A directed intersection representation is an assignment of a color set to each vertex in a digraph such that two vertices form an edge if and only if their color sets share at least one color and the tail vertex has a strictly smaller color set than the head. The smallest possible size of the union of the color sets is defined to be the directed intersection number (DIN). We show that the directed intersection representation is well-defined for all directed acyclic graphs and the maximum DIN among all n vertex acyclic digraphs is at most 5n^2/8 + O(n) and at least 9n^2/16 + O(n)
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