Rainbow Subgraphs and their Applications

We consider the Minimum Rainbow Subgraph problem (MRS): Given a graph G , whose edges are coloured with p colours. Find a subgraph F ⊆ G of minimum order and with p edges such that each colour occurs exactly once. This problem is NP-hard and APX-hard. For a given graph G and an edge colouring c with p colours we define the rainbow subgraph number rs ( G , c ) to be the order of a minimum rainbow subgraph of G with size p . In this paper we will show lower and upper bounds for the rainbow subgraph number of a graph.

The NP-hard RAINBOW SUBGRAPH problem, motivated from bioinformatics, is to find in an edge-colored graph a subgraph that contains each edge color exactly once and has at most \(k\) vertices. We examine the parameterized complexity of RAINBOW SUBGRAPH for paths, trees, and general graphs. We show that RAINBOW SUBGRAPH is W[1]-hard with respect to the parameter \(k\) and also with respect to the dual parameter \(\ell:=n-k\) where \(n\) is the number of vertices. Hence, we examine parameter combinations and show, for example, a polynomial-size problem kernel for the combined parameter \(\ell\) and ``maximum number of colors incident with any vertex''. Additionally, we show APX-hardness even if the input graph is a properly edge-colored path in which every color occurs at most twice.

The NP-hard Rainbow Subgraph problem, motivated from bioinformatics, is to find in an edge-colored graph a subgraph that contains each edge color exactly once and has at most \(k\) vertices. We examine the parameterized complexity of Rainbow Subgraph for paths, trees, and general graphs. We show, for example, APX-hardness even if the input graph is a properly edge-colored path in which every color occurs at most twice. Moreover, we show that Rainbow Subgraph is W[1]-hard with respect to the parameter \(k\) and also with respect to the dual parameter \(\ell :=n-k\) where \(n\) is the number of vertices. Hence, we examine parameter combinations and show, for example, a polynomial-size problem kernel for the combined parameter \(\ell \) and “maximum number of colors incident with any vertex”.

The general structure of colored complete graphs containing no copy of a particular rainbow subgraph has been extremely useful in establishing sharp Ramsey-type results for finding monochromatic subgraphs. Several small graphs, like \(P_{3}\) for example, immediately trivialize the problem. Indeed, if a colored complete graph contains no rainbow copy of \(P_{3}\), then it must be colored entirely with one color. Adding in the third edge to make a triangle already makes the problem much more interesting. When a rainbow subgraph G is forbidden from a coloring, we say the coloring is rainbow G-free . For example, if a rainbow triangle is forbidden, we say the coloring is rainbow triangle-free .

Rainbow edge-coloring and rainbow domination

Improved bounds for anti-Ramsey numbers of matchings in outer-planar graphs

We consider a canonical Ramsey type problem. An edge‐coloring of a graph is called m‐good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m‐good edge‐coloring of Kn yields a properly edge‐colored copy of G, and let g(m, G) denote the smallest n such that every m‐good edge‐coloring of Kn yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = Kt, we have c1mt2/ln t ≤ f(m, Kt) ≤ c2mt2, and cmt3/ln t ≤ g(m, Kt) ≤ cmt3/ln t, where c1, c2, c, c are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003

Implications in rainbow forbidden subgraphs

A rainbow subgraph in an edge-coloured graph is a subgraph such that its edges have distinct colours. The minimum colour degree of a graph is the smallest number of distinct colours on the edges incident with a vertex over all vertices. Kostochka, Pfender, and Yancey showed that every edge-coloured graph on n vertices with minimum colour degree at least k contains a rainbow matching of size at least k, provided $${n\geq \frac{17}{4}k^2}$$ . In this paper, we show that n ≥ 4k − 4 is sufficient for k ≥ 4.

An edge‐coloring of a complete graph with a set of colors is called completely balanced if any vertex is incident to the same number of edges of each color from . Erdős and Tuza asked in 1993 whether for any graph on edges and any completely balanced coloring of any sufficiently large complete graph using colors contains a rainbow copy of . This question was restated by Erdős in his list of “Some of my favourite problems on cycles and colourings.” We answer this question in the negative for most cliques by giving explicit constructions of respective completely balanced colorings. Further, we answer a related question concerning completely balanced colorings of complete graphs with more colors than the number of edges in the graph .

The rainbow number of matchings in regular bipartite graphs

We consider quadruples of positive integers with and such that every proper edge-coloring of the complete bipartite graph contains a rainbow subgraph. We show that every such quadruple with and satisfies this property and find an infinite sequence where this bound is sharp. We also define and compute some new anti-Ramsey numbers.

Long directed rainbow cycles and rainbow spanning trees

We study graph theory and combinatorics which are topics in discrete mathematics. The graphs we consider in the thesis consist of a set of vertices and a set of edges in which every edge joins two vertices. An edge-coloring of a graph is an assignment of colors to the edges of the graph.One fundamental problem in the research of edge-colored graphs is to study the existence of nice substructures in an edge-colored host graph. In this thesis, the nice substructure we consider is either a rainbow subgraph or a monochromatic subgraph, and the host graph is a complete graph. For two graphs G and H, the k-colored Gallai-Ramsey number is the minimum integer n such that every k-edge-coloring of the complete graph on n vertices contains either a rainbow copy of G or a monochromatic copy of H. This concept can be considered as a generalization of the classical Ramsey number.In this thesis, we determine the exact values of the Gallai-Ramsey numbers for rainbow triangles and several monochromatic subgraphs. We also obtain some exact values and bounds for the Ramsey numbers and Gallai-Ramsey numbers of a class of unicyclic graphs. In addition, we contribute some new results related to this research area. In particular, we study an extremal problem related to Gallai-colorings, the Gallai-Ramsey multiplicity problem, the Erdős-Gyárfás function with respect to Gallai-colorings, the forbidden rainbow subgraph condition for the existence of a highly-connected monochromatic subgraph, and the rainbow Erdős-Rothschild problem with respect to 3-term arithmetic progressions.Throughout this thesis, we present several open problems and conjectures that remain unsolved. In particular, a driving problem is to determine the Gallai-Ramsey numbers for complete graphs. This problem is related to the classical 2-colored Ramsey numbers for complete graphs, and also has a close relationship with the multicolor Erdős-Hajnal conjecture. Another important problem is to study how does the additional constraint on rainbow triangles influence the classical extremal problems. We hope that these problems and conjectures attract more attention from other researchers.

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