Abstract
We show that the diameter of connected $k$-colorable graphs with minimum degree $\geq \delta$ and order $n$ is at most $\left(3-\frac{1}{k-1}\right)\frac{n}{\delta}-1$, while for $k=3$, it is at most $\frac{57n}{23\delta}+O\left(1\right)$.
Highlights
We show that the diameter of connected k-colorable graphs with minimum degree δ and order n is at most
This paper is concerned with the maximum diameter of connected graphs, namely how it depends on the order and the minimum degree, and possibly on further graph properties
Let r, δ 2 be fixed integers and let G be a connected graph of order n and minimum degree δ
Summary
This paper is concerned with the maximum diameter of connected graphs, namely how it depends on the order and the minimum degree, and possibly on further graph properties. − 1, if G is a Kk+1-free (in a weaker version of the conjecture: k-colorable) connected graph of order n and minimum degree at least δ, diam(G). At proving upper bounds on the diameter, we can assume that those canonical properties hold. If G is a connected k-colorable graph of minimum degree at least δ 1, This corroborates the conjecture of Erdos et al in the sense that the maximum diameter among all graphs investigated in Theorem 5 is 3−Θ.
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