Auswahl der wissenschaftlichen Literatur zum Thema „Rainbow subgraph“

A subgraph H in an edge-colouring is properly coloured if incident edges of H are assigned different colours, and H is rainbow if no two edges of H are assigned the same colour. We study properly coloured subgraphs and rainbow subgraphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.

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.

Arainbow subgraphof an edge-coloured graph is a subgraph whose edges have distinct colours. Thecolour degreeof a vertexvis the number of different colours on edges incident withv. Wang and Li conjectured that fork≥ 4, every edge-coloured graph with minimum colour degreekcontains a rainbow matching of size at least ⌈k/2⌉. A properly edge-colouredK4has no such matching, which motivates the restrictionk≥ 4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size ⌊k/2⌋ is guaranteed to exist, and they proved several sufficient conditions for a matching of size ⌈k/2⌉. We prove the conjecture in full.

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. 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(st2) 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 paper, we study this case and show that f(S, Pt) = 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.

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.

Dans cette thèse, nous étudions les sous graphes arc-en-ciel et les sous-graphes correctement colorés dans les graphes à arêtes colorées, et les sous-graphes compatibles dans les graphes avec des systèmes d'incompatibilité, qui peuvent être considérés comme une généralisation des graphes à arêtes colorées. Par rapport aux graphes généraux, les graphes colorés contiennent plus d'informations et sont capables de modéliser des relations plus complexes dans les réseaux de communication, les sciences sociales, la biologie moléculaire, etc. Par conséquent, l'étude des structures dans les graphes aux arêtes colorées est importante à la fois pour la théorie des graphes et pour d'autres sujets connexes. Nous étudions d'abord la condition de degré de couleur minimum forçant les triangles arc-en-ciel à sommets disjoints dans les graphes aux arêtes colorées. En 2013, Li s'est avéré être la meilleure condition de degré de couleur minimum possible pour l'existence d'un triangle arc-en-ciel. Motivés par cela, nous obtenons une condition de degré de couleur minimum précis garantissant l'existence de deux triangles arc-en-ciel à sommets disjoints et proposons une conjecture sur l'existence de k triangles arc-en-ciel à sommets disjoints. Deuxièmement, nous considérons la relation entre l'ordre de l'arbre maximum correctement coloré dans le graphe à bords colorés et le degré de couleur minimum. On obtient que pour un graphe connexe G aux arêtes colorées, l'ordre du maximum d'arbre correctement coloré est au moins \min\{|G|, 2\delta^{c}(G)\}, ce qui généralise un résultat de Cheng, Kano et Wang. De plus, la borne inférieure 2delta^{c}(G) dans notre résultat est la meilleure possible et nous caractérisons tous les graphes extrémaux. Troisièmement, nous recherchons la condition de degré de couleur minimum garantissant l'existence de 2-facteurs correctement colorés dans les graphes aux bords colorés. Nous dérivons une condition de degré de couleur minimum asymptotique forçant chaque facteur 2 correctement coloré avec exactement t composants, ce qui généralise un résultat de Lo. Nous déterminons également la meilleure condition de degré de couleur minimum possible pour l'existence d'un facteur 2 correctement coloré dans un graphe bipartite à arêtes colorées. Enfin, nous étudions les facteurs compatibles dans les graphes avec des systèmes d'incompatibilité. La notion de système d'incompatibilité a été introduite pour la première fois par Krivelevich, Lee et Sudakov, qui peut être considérée comme une mesure quantitative de la robustesse des propriétés du graphe. Récemment, il y a eu un intérêt croissant pour l'étude de la robustesse des propriétés des graphes, visant à renforcer les résultats classiques en théorie des graphes extrémaux et en combinatoire probabiliste. Nous étudions la version robuste du résultat d'Alon-- Yuster par rapport au système d'incompatibilité
In this thesis, we study rainbow subgraphs and properly colored subgraphs in edge-colored graphs, and compatible subgraphs in gra-phs with incompatibility systems, which can be viewed as a generalization of edge-colored graphs. Compared with general graphs, edge-colored gra-phs contain more information and are able to model more complicated relations in communication net-work, social science, molecular biology and so on. Hence, the study of structures in edge-colored graphs is significant to both graph theory and other related subjects. We first study the minimum color degree condition forcing vertex-disjoint rainbow triangles in edge-colored graphs. In 2013, Li proved a best possible minimum color degree condition for the existence of a rainbow triangle. Motivated by this, we obtain a sharp minimum color degree condition guaran-teeing the existence of two vertex-disjoint rainbow triangles and propose a conjecture about the exis-tence of k vertex-disjoint rainbow triangles. Secondly, we consider the relation between the order of maximum properly colored tree in edge-colored graph and the minimum color degree. We obtain that for an edge-colored connected graph G, the order of maximum properly colored tree is at least \min\{|G|, 2\delta^{c}(G)\}, which generalizes a result of Cheng, Kano and Wang. Moreover, the lower bound 2delta^{c}(G) in our result is best possible and we characterize all extremal graphs. Thirdly, we research the minimum color degree condition guaranteeing the existence of properly colored 2-factors in edge-colored graphs. We derive an asymptotic minimum color degree con-dition forcing every properly colored 2-factor with exactly t components, which generalizes a result of Lo. We also determine the best possible mini-mum color degree condition for the existence of a properly colored 2-factor in an edge-colored bipartite graph. Finally, we study compatible factors in graphs with incompatibility systems. The notion of incom-patibility system was firstly introduced by Krivelevich, Lee and Sudakov, which can be viewed as a quantitative measure of robustness of graph properties. Recently, there has been an increasing interest in studying robustness of graph proper-ties, aiming to strengthen classical results in extremal graph theory and probabilistic combina-torics. We study the robust version of Alon--Yuster's result with respect to the incompatibility system