By Xu B.G.

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Small transmission delay, that is, small diameter or average distance. Since transmission delay or signal degradation for sending a message from one vertex to another is approximately proportional to the number of times that a message has to be stored and forwarded by intermediate vertices. Thus, a small average distance or diameter is desired to obtain a highly efficient interconnection network. In particular, diameter should be bounded by a given value for a real-time processing system. 3. Maximum fault tolerance.

1. LINE GRAPHICAL METHOD (a) L(G) is simple and II(L(G)) = c(G). (b) dL(G)(e) = dG(x) + dG(Y) - 2 for any e = xy E E(G), and hence 8(L(G)) = ~(G). In particular, L(G) is (2d - 2)-regular if Gis d-regular. (c) For any x E V (G), the subgraph of L (G) induced by the edges incident with x E V(G) is a complete graph. (d) c(L(G)) =~ (dG(x))2 - c(G). ~ xEV(G) (e) For a connected undirected graph G, L(G) undirected cycle. ~ G if and only if G is an Proof It is a quite simple observation to prove the assertions (a), (b) and (c) by the definition of the line graph.

We use Oc and h to denote the minimum degree of G and L, respectively. Use AC and AL to denote the edge-connectivity of G and L, respectively. (G) + o(G) - 2, where ~(G) is the minimum edge-degree of G. To state the results in this subsection, we would like to introduce some notation. 1. LINE GRAPHICAL METHOD Let B C V(L) = E(G) and B = V(L) \ B such that EL(B) is a AL-cut of L. For any x E V(G), let = EL(B, B) = IEa(x) n BI, and p(x) = IEa(x) n BI. da(x) = p(x) + p(x). Let X = {x E V(G): p(x) > 0 and p(x) > O}.

### A 3-color Theorem on Plane Graphs without 5-circuits by Xu B.G.

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