By Frank Harary, Robert Z. Norman, Dorwin Cartwright

ISBN-10: 047135130X

ISBN-13: 9780471351306

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**Sample text**

Since the x's are distinct and u = *ο, it can only be that e = [xo,x\}. , xr = wj] is a path in G — e from v to w2. Hence, v e Η2· A similar argument now shows that u € H\. Because u and v are in different components of G — e, edge e must lie along every path in G from u to w. , yt = v] is such a path, then, since the ;y's are distinct and e = uv is an edge of P, we have v = y\\ that is, k = 1 and P — [u, v]. 3 that a cut-edge is sometimes called a bridge. (See Fig. ) If e(G) = 1, then G is barely connected in the sense that it can be disconnected by the removal of a single, well-chosen edge.

Let u € V(G) be fixed but arbitrary. Suppose w e V(G). If w = u or d(u, w) is even, color vertex w blue; if d(u, w) is odd, color it green. This fails to produce a proper 2-coloring of G only if there exist adjacent vertices w\ and M>2 that are colored the same. , ys] be a shortest path from u = yo to w2 = ys. If {ΛΓ,: 1 < i < r] Π {y;: 1 < j < s] φ φ, let k be the largest index such that Xk € {;y,: 1 < j < s}. Because P and Q are shortest paths, it must be that xk = yk and (JC,:fc< / < r} Π [yj\ k < j < s} = φ.

Without computing p(G,x), show that it could not possibly be equal to f(x) = p(C4,x)2/p(Pi,x). Prove that p(C„, JC) = (x - 1 )" + ( - 1 )" (JC - 1 ). 27 The wheel o n n + 1 vertices is W„ = C„ v K\. ) Compute p(Wn,x). 28 Suppose G is a fixed but arbitrary graph. Prove or disprove that the roots of p(G, x) are all real. x is not the chromatic polynomial of any graph. 30 Let G = (V, E) be a graph with n vertices and m edges. Suppose e = uv € E. To subdivide e means, informally, to put a new vertex in the middle of e.

### Structural Models: An Introduction to the Theory of Directed Graphs by Frank Harary, Robert Z. Norman, Dorwin Cartwright

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