By Frank Harary

Awarded in 1962–63 through specialists at college university, London, those lectures supply a number of views on graph conception. even though the hole chapters shape a coherent physique of graph theoretic recommendations, this quantity isn't a textual content at the topic yet really an creation to the vast literature of graph concept. The seminar's themes are aimed toward complicated undergraduate scholars of mathematics.

Lectures by way of this volume's editor, Frank Harary, contain "Some Theorems and ideas of Graph Theory," "Topological suggestions in Graph Theory," "Graphical Reconstruction," and different introductory talks. a sequence of invited lectures follows, that includes shows through different professionals at the college of college university in addition to traveling students. those contain "Extremal difficulties in Graph idea" by way of Paul Erdös, "Complete Bipartite Graphs: Decomposition into Planar Subgraphs," by means of Lowell W. Beineke, "Graphs and Composite Games," via Cedric A. B. Smith, and a number of other others.

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The first purpose of the publication is to introduce an array of lovely difficulties in quite a few matters speedy, pithily and entirely carefully to graduate scholars and complex undergraduates. The booklet takes a couple of particular difficulties and solves them, the wanted instruments constructed alongside the best way within the context of the actual difficulties.

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**Example text**

Now the proof follows by a simple probabilistic argument. Namely, we color every edge e ∈ E by choosing one of the colors in K = {a1 , · · · , a4 , b1 , · · · , b4 } uniformly and independently at random. Observe that a fixed path P of length at most four is an a-rainbow path with probability at least 8−4 . Similarly, P is a b-rainbow path with probability at least 8−4 . So any fixed pair u, v ∈ Vi is 32 3 Upper Bounds for Rainbow Connection Numbers not both a-rainbow-connected and b-rainbow-connected with probability at most 5 2(1 − 8−4 )8 log n < n−2 , and therefore the probability that all such pairs are both a-rainbow connected and b-rainbow connected is strictly positive.

The above bound may not be tight, and we are tempted to believe that the following conjecture might be true. 12 ([14, 71]). For every κ ≥ 1, if G is a κ -connected graph of order n, then rc(G) ≤ n/κ + C, where C is a constant. 12 is true, namely high girth graphs, chordal graphs. 13 ([14,71]). If G is a κ -connected graph such that κ ≥ 3 and g(G) ≥ 7, then rc(G) ≤ n/κ + 41. If G is κ -connected such that κ ≥ 5 and g(G) ≥ 5, then rc(G) ≤ n/κ + 19. Actually, the above result holds when replacing κ by δ .

We will first consider an intermediate problem called the k-subset strong rainbow connected problem which is the decision version of the subset strong rainbow connected problem. The input to the k-subset strong rainbow connected problem is a graph G along with a set of pairs P = {(u, v) : (u; v) ⊆ V × V } and an integer k. Our goal is to answer whether there exists an edge-coloring of G with at most k colors such that every pair (u, v) ∈ P has a geodesic rainbow path. Let G = (V, E) be an instance of the k-vertex-coloring problem.

### A Seminar on Graph Theory by Frank Harary

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