By S.Francinou, H.Gianella, S.Nicolas
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Extra resources for Exercices de mathématiques Oraux de l'ENS : Algebre 2
Conjecture of Berge-Duchet ). Is it true that (7) j (8)? (Perfect Graph Conjecture). Is it true that (6) j (8)? (Weak form of the Perfect Graph Conjecture). Is it true that the odd circuits are the only connected kernless graphs such that the removal of any arc results in a graph with a kernel? (Conjecture of Duchet, [91) References [l] C. Berge, Graphs, North-Holland Mathematical Library, Vol. 6 (North-Holland, Amsterdam, 1985) Chapter 14.  C. Berge, Nouvelles extensions du noyau d‘un graphe et ses applications en theorie des jeux, Publ.
The graph H obtained by removing two adjacent edges from C,, consists of an isolated vertex and a path of order n - 1. Thus, a ( H ) = 1 + a(P,,-,) = 1 + [(n - 1)/3] = 1 + [n/3] = 1 + a(C,,), whence b(C,,)S 2 in this case. Combining this with the upper bound obtained earlier, we have b(C,,) = 2 if n = 0, 2 (mod 3). Case 2. Suppose now that n = 1(mod 3). The graph H resulting from the deletion of three consecutive edges of C,, consists of two isolated vertices and a path of order n - 2. Thus, + [ ( n - 2)/3] = 2 + (n - 1)/3 = 2 + ( I d 3 1 - 1) = 1 + ~(c,,), a ( H )= 2 so that b(C,,)S 3.
If b ( G )S deg u, then b (G) s p - 1, so we suppose that b (G) > deg u. Let Eu denote the set of edges incident with u. Then a(G - E,) = a(G) and a(G - u ) = a(G) - 1. Also, if D denotes the union of all minimum dominating sets for G - u , then u is adjacent in G to no vertex of D . Hence, JE,(s p - 1 - (Dl and v t$ D . Now, if F, denotes the set of edges from v to a vertex in D , then since u t$ D we must have a(G - u - F , ) > a(G - u ) , or equivalently, a(G - u - F , ) > a(G) - 1. Thus a((; - (E, U 4))> a(G) and we see that b ( G )S IEU u F,I = lEUl+ lFvl S ( p - 1 - IDl)+ ID1 = p - 1.