Long properly colored cycles in edge colored complete graphs

Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let $\Delta^{\mathrm{mon}} (K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K_{n}^{c}$. A properly colored cycle (path) in $K_{n}^{c}$ is a cycle (path) in which adjacent edges have distinct colors. B. Bollob\'{a}s and P. Erd\"{o}s (1976) proposed the following conjecture: if $\Delta^{\mathrm{mon}} (K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if $\Delta^{\mathrm{mon}} (K_{n}^{c})< \lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored cycle of length at least $\lceil \frac{n+2}{3}\rceil+1$. In this paper, we improve the bound to $\lceil \frac{n}{2}\rceil + 2$.Comment: 8 page

Extending perfect matchings to Hamiltonian cycles in line graphs

A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a Hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH-property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most $3$, (ii) a complete graph, or (iii) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.Comment: 12 pages, 4 figure

Rainbow Turán Problems

For a fixed graph H, we define the rainbow Turán number ex^*(n,H) to be the maximum number of edges in a graph on n vertices that has a proper edge-colouring with no rainbow H. Recall that the (ordinary) Turán number ex(n,H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H. For any non-bipartite H we show that ex^*(n,H)=(1+o(1))ex(n,H), and if H is colour-critical we show that ex^{*}(n,H)=ex(n,H). When H is the complete bipartite graph K_{s,t} with s ≤ t we show ex^*(n,K_{s,t}) = O(n^{2-1/s}), which matches the known bounds for ex(n,K_{s,t}) up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound ex^*(n,C_6) = O(n^{4/3}), which is of the correct order of magnitude

Properly Edge-colored Theta Graphs in Edge-colored Complete Graphs

With respect to specific cycle-related problems, edge-colored graphs can be considered as a generalization of directed graphs. We show that properly edge-colored theta graphs play a key role in characterizing the difference between edge-colored complete graphs and multipartite tournaments. We also establish sufficient conditions for an edge-colored complete graph to contain a small and a large properly edge-colored theta graph, respectively

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

Circuits of edge-coloured complete graphs

This thesis investigates the circuits of edge-coloured complete graphs. There are various kinds of edge-coloured circuits. Amongst the most interesting are polychromatic circuits (each edge is differently coloured), alternating circuits (adjacent edges are differently coloured) and monochromatic circuits (every edge is the same colour). In the case of triangles, there are monochromatic, bichromatic (2-edge-coloured) and polychromatic triangles. This thesis is an investigation into edge-coloured complete graphs which do not contain one of the above kinds of circuits. Special emphasis is given to those edge-coloured complete graphs which do not contain one type of triangle, and those in which two types of triangle are forbidden. Where possible, the structure of these graphs is determined and a method of construction given; various types of extremal results are obtained