We study several extremal problems in graph labelling and in weak diameter of digraphs.
In Chapter 2 we apply the Discharging Method to prove the 1,2,3-Conjecture [41] and the
1,2-Conjecture [48] for graphs with maximum average degree less than 8/3. Stronger results on
these conjectures have been proved, but this is the first application of discharging to them,
and the structure theorems and reducibility results are of independent interest. Chapter 2
is based on joint work with D. Cranston and D. West that appears in [17].
In Chapter 3 we focus on digraphs. The weak distance between two vertices x and y in a
digraph G is the length of the shortest directed path from x to y or from y to x. We define
the weak diameter of a digraph to be the maximum directed distance among all pairs of
vertices of the digraph. For a fixed integer D, we determine the minimum number of edges
in a digraph with weak diameter at least D, when D = 2, or when the number of vertices of
the digraph is very large or small with respect to D. Chapter 3 is based on joint work with
Z. Furedi that appears in [26].
In Chapter 4 using Ramsey graphs, we determine the minimum clique size an n-vertex
graph with chromatic number \chi can have if \chi \geq (n+3)/2. For integers n and t, we determine
the maximum number of colors in an edge-coloring of a complete graph Kn that does not
have t edge-disjoint rainbow spanning trees of Kn. For integers t and n, we also determine
the maximum number of colors in an edge-coloring of Kn that does not have any rainbow
spanning subgraph with diameter t. Chapter 4 is based on three papers, the first is joint
work with C. Biro and Z. Furedi [11] and the other two are joint work with D. West [36, 37]
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