Covering a cubic graph with perfect matchings

Let G be a bridgeless cubic graph. A well-known conjecture of Berge and Fulkerson can be stated as follows: there exist five perfect matchings of G such that each edge of G is contained in at least one of them. Here, we prove that in each bridgeless cubic graph there exist five perfect matchings covering a portion of the edges at least equal to 215/231 . By a generalization of this result, we decrease the best known upper bound, expressed in terms of the size of the graph, for the number of perfect matchings needed to cover the edge-set of G.Comment: accepted for the publication in Discrete Mathematic

Berge - Fulkerson Conjecture And Mean Subtree Order

Let $G$ be a graph, $V (G)$ and $E(G)$ be the vertex set and edge set of $G$, respectively. A perfect matching of $G$ is a set of edges, $M\subseteq E(G)$, such that each vertex in $G$ is incident with exactly one edge in $M$. An $r$-regular graph is said to be an $r$-graph if $|\partial(X)| \geq r$ for each odd set $X \subseteq V(G)$, where $|\partial(X)|$ denotes the set of edges with precisely one end in $X$. One of the most famous conjectures in Matching Theory, due to Berge, states that every 3-graph $G$ has five perfect matchings such that each edge of $G$ is contained in at least one of them. Likewise, generalization of the Berge Conjecture given, by Seymour, asserts that every $r$-graph $G$ has $2r-1$ perfect matchings that covers each $e \in E(G)$ at least once. In the first part of this thesis, I will provide a lower bound to number of perfect matchings needed to cover the edge set of an $r$-graph. I will also present some new conjectures that might shade a light towards the generalized Berge conjecture. In the second part, I will present a proof of a conjecture stating that there exists a pair of graphs $G$ and $H$ with $H\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that mean subtree order of $H$ is smaller then mean subtree order of $G$

Fractional total colourings of graphs of high girth

Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and for even Delta>=4 in the following stronger form: For each of these values of Delta, there exists g such that the fractional total chromatic number of any graph with maximum degree Delta and girth at least g is equal to Delta+1

Circuits, Perfect Matchings and Paths in Graphs

We primarily consider the problem of finding a family of circuits to cover a bidgeless graph (mainly on cubic graph) with respect to a given weight function defined on the edge set. The first chapter of this thesis is going to cover all basic concepts and notations will be used and a survey of this topic.;In Chapter two, we shall pay our attention to the Strong Circuit Double Cover Conjecture (SCDC Conjecture). This conjecture was verified for some graphs with special structure. As the complement of two factor in cubic graph, the Berge-Fulkersen Conjecture was introduced right after SCDC Conjecture. In Chapter three, we shall present a series of conjectures related to perfect matching covering and point out their relationship.;In last chapter, we shall introduce the saturation number, in contrast to extremal number (or known as Turan Number), and describe the edge spectrum of saturation number for small paths, where the spectrum was consisted of all possible integers between saturation number and Turan number

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

Topics in graph colouring and graph structures

This thesis investigates problems in a number of different areas of graph theory. These problems are related in the sense that they mostly concern the colouring or structure of the underlying graph. The first problem we consider is in Ramsey Theory, a branch of graph theory stemming from the eponymous theorem which, in its simplest form, states that any sufficiently large graph will contain a clique or anti-clique of a specified size. The problem of finding the minimum size of underlying graph which will guarantee such a clique or anti-clique is an interesting problem in its own right, which has received much interest over the last eighty years but which is notoriously intractable. We consider a generalisation of this problem. Rather than edges being present or not present in the underlying graph, each is assigned one of three possible colours and, rather than considering cliques, we consider cycles. Combining regularity and stability methods, we prove an exact result for a triple of long cycles. We then move on to consider removal lemmas. The classic Removal Lemma states that, for n sufficiently large, any graph on n vertices containing o(n^3) triangles can be made triangle-free by the removal of o(n^2) edges. Utilising a coloured hypergraph generalisation of this result, we prove removal lemmas for two classes of multinomials. Next, we consider a problem in fractional colouring. Since finding the chromatic number of a given graph can be viewed as an integer programming problem, it is natural to consider the solution to the corresponding linear programming problem. The solution to this LP-relaxation is called the fractional chromatic number. By a probabilistic method, we improve on the best previously known bound for the fractional chromatic number of a triangle-free graph with maximum degree at most three. Finally, we prove a weak version of Vizing's Theorem for hypergraphs. We prove that, if H is an intersecting 3-uniform hypergraph with maximum degree D and maximum multiplicity m, then H has at most 2D+m edges. Furthermore, we prove that the unique structure achieving this maximum is m copies of the Fano Plane