Large rainbow matchings in large graphs

A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough (namely, $n\geq 4.25k^2$), then each $n$-vertex graph $G$ with minimum color degree at least $k$ contains a rainbow matching of size at least $k$

BILANGAN KETERHUBUNGAN PELANGI PADA PEWARNAAN-SISI GRAF

Let  be a graph. An edge-coloring of  is a function , where  is a set of colors. Respect to  a subgraph  of  is called a rainbow subgraph if all edges of  get different colors. Graph  is called rainbow connected if for every two distinct vertices of  is joined by a rainbow path. The rainbow connection number of , denoted by , is the minimum number of colors needed in coloring all edges of  such that  is a rainbow connected. The main problem considered in this thesis is determining the rainbow connection number of graph. In this thesis, we determine the exact value of the rainbow connection number of some classes of graphs such as Cycles, Complete graph, and Tree. We also determining the lower bound and upper bound for the rainbow connection number of graph. Keywords: Rainbow Connection Number, Graph, Edge-Coloring on Graph. &nbsp

Existences of rainbow matchings and rainbow matching covers

Let $G$ be an edge-coloured graph. A rainbow subgraph in $G$ is a subgraph such that its edges have distinct colours. The minimum colour degree $\delta^c(G)$ of $G$ is the smallest number of distinct colours on the edges incident with a vertex of $G$. We show that every edge-coloured graph $G$ on $n\geq 7k/2+2$ vertices with $\delta^c(G) \geq k$ contains a rainbow matching of size at least $k$, which improves the previous result for $k \ge 10$. Let $\Delta_{\text{mon}}(G)$ be the maximum number of edges of the same colour incident with a vertex of $G$. We also prove that if $t \ge 11$ and $\Delta_{\text{mon}}(G) \le t$, then $G$ can be edge-decomposed into at most $\lfloor tn/2 \rfloor$ rainbow matchings. This result is sharp and improves a result of LeSaulnier and West

Colored Saturation Parameters for Bipartite Graphs

Let F and H be fixed graphs and let G be a spanning subgraph of H. G is an F-free subgraph of H if F is not a subgraph of G. We say that G is an F-saturated subgraph of H if G is F-free and for any edge e in E(H)-E(G), F is a subgraph of G+e. The saturation number of F in K_{n,n}, denoted sat(K_{n,n}, F), is the minimum size of an F-saturated subgraph of K_{n,n}. A t-edge-coloring of a graph G is a labeling f: E(G) to [t], where [t] denotes the set { 1, 2, ..., t }. The labels assigned to the edges are called colors. A rainbow coloring is a coloring in which all edges have distinct colors. Given a family F of edge-colored graphs, a t-edge-colored graph H is (F, t)-saturated if H contains no member of F but the addition of any edge in any color completes a member of F. In this thesis we study the minimum size of ( F,t)-saturated subgraphs of edge-colored complete bipartite graphs. Specifically we provide bounds on the minimum size of these subgraphs for a variety of families of edge-colored bipartite graphs, including monochromatic matchings, rainbow matchings, and rainbow stars

Generalized Colorings of Graphs

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

Constrained Ramsey Numbers

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision

Introduction to the Minimum Rainbow Subgraph problem

Arisen from the Pure Parsimony Haplotyping problem in the bioinformatics, we developed the Minimum Rainbow Subgraph problem (MRS problem): Given a graph $G$, whose edges are coloured with $p$ colours. Find a subgraph $F\\\\subseteq G$ of $G$ of minimum order and with $p$ edges such that each colour occurs exactly once. We proved that this problem is NP-hard, and even APX-hard. Furthermore, we stated upper and lower bounds on the order of such minimum rainbow subgraphs. Several polynomial-time approximation algorithms concerning their approximation ratio and complexity were discussed. Therefore, we used Greedy approaches, or introduced the local colour density $\\\\lcd(T,S)$, giving a ratio on the number of colours and the number of vertices between two subgraphs $S,T\\\\subseteq G$ of $G$. Also, we took a closer look at graphs corresponding to the original haplotyping problem and discussed their special structure.:Mathematics and biology - having nothing in common? I. Going for a start 1. Introducing haplotyping 2. Becoming mathematical II. The MRS problem 3. The graph theoretical point of view 3.1. The MRS problem 3.2. The MRS problem on special graph classes 4. Trying to be not that bad 4.1. Greedy approaches 4.2. The local colour density 4.3. MaxNewColour 5. What is real data telling us? And the work goes on and on Bibliograph