Partitioning 2-edge-coloured bipartite graphs into monochromatic cycles

Long monochromatic paths and cycles in 2-colored bipartite graphs

Monochromatic cycles in 2-edge-colored bipartite graphs with large minimum degree

Partitioning Complete Bipartite Graphs by Monochromatic Cycles

Graph partitioning problems enjoy many practical applications as well as algorithmic and theoretical challenges. This motivates the topics of this thesis that is composed of two parts. The first part of the thesis consists of Chapters 2 to 4. In this part, we present results on the complexity and inapproximability of some vertex partitioning problems, and we give approximation algorithms and on-line algorithms for some other vertex partitioning problems. We will start by investigating the inapproximability and complexity of the problems of finding the minimum number of monochromatic cliques and rainbow cycles that, respectively, partition V (G), where the graph G avoids some forbidden induced subgraphs. Secondly, we study the complexity, and develop approximation algorithms and on-line algorithms for injective coloring problems, to be defined later. Finally, we consider the design of a semidefinite programming based approximation algorithm for a variant of the max hypergraph cut problem. The second part of the thesis consists of Chapters 5 to 7. In this part, we turn our attention to structural properties of some problems that are related to matching problems which be regarded as edge partitioning problems. Firstly, we determine the minimum size of a k-extendable bipartite graph and that of an n-factor-critical graph, and we study the problem of determining the minimum size of a k-extendable non-bipartite graph. We solve this problem for k = 1 and k = 2, and we pose a conjecture related to the problem for general k. Secondly, we improve two equivalent structural results due to Woodall and Las Vergnas on the existence of a directed Hamilton cycle in a digraph and the containment of every perfect matching in a Hamilton cycle in a balanced (undirected) bipartite graph, respectively. Finally, we study a generalization of the maximum matching problem called the maximum triangle set problem, in which the aim is to find the maximum number of vertex-disjoint triangles of a given graph. We present a necessary and sufficient condition for augmenting triangle sets, analogous to the well-known augmenting path result for matchings.

We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of S\'ark\"ozy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2\alpha(G)$ monochromatic cycles, where $\alpha(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1+\eps){3|V(G)|\over 4}$ can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that $\overline{G}$ does not contain a fixed bipartite graph $H$, we show that in every $2$-edge-coloring of $G$, $|V(G)|-c(H)$ vertices can be covered by two vertex disjoint paths of different colors, where $c(H)$ is a constant depending only on $H$. In particular, we prove that $c(C_4)=1$, which is best possible.

We answer a question of Gyárfás and Sárközy from 2013 by showing that every 2-edge-coloured complete 3-uniform hypergraph can be partitioned into two monochromatic tight paths of different colours. We also give a lower bound for the number of tight paths needed to partition any 2-edge-coloured complete k-partite k-uniform hypergraph. Finally, we show that any 2-edge coloured complete bipartite graph has a partition into a monochromatic cycle and a monochromatic path, of different colours, unless the colouring is a split colouring.

In this paper we consider the problem of partitioning complete multipartite graphs with edges colored by 2 colors into the minimum number of vertex disjoint monochromatic cycles, paths and trees, respectively. For general graphs we simply address the decision version of these three problems the 2-PGMC, 2-PGMP and 2-PGMT problems, respectively. We show that both 2-PGMC and 2-PGMP problems are NP-complete for complete multipartite graphs and the 2-PGMT problem is NP-complete for bipartite graphs. This also implies that all these three problems are NP-complete for general graphs, which solves a question proposed by the authors in a previous paper. Nevertheless, we show that the 2-PGMT problem can be solved in polynomial time for complete multipartite graphs.

Partitioning 3-edge-coloured complete bipartite graphs into monochromatic cycles

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

Local colourings and monochromatic partitions in complete bipartite graphs

Local colourings and monochromatic partitions in complete bipartite graphs

Assume that the edges of a complete bipartite graph K(A, B) are colored with r colors. In this paper we study coverings of B by vertex disjoint monochromatic cycles, connected matchings, and connected subgraphs. These problems occur in several applications.

We show that for any coloring of the edges of the complete bipartite graph $K_{n,n}$ with three colors there are five disjoint monochromatic cycles which together cover all but $o(n)$ of the vertices. In the same situation, 18 disjoint monochromatic cycles together cover all vertices.