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**1 - 2**of**2**### The sum of degrees in cliques

"... For every graph G, let ∆r (G) =max d (u):R is an r-clique of G u∈R and let ∆r (n, m) be the minimum of ∆r (G) taken over all graphs of order n and size m. Writetr (n) forthesizeofther-chromatic Turán graph of order n. Improving earlier results of Edwards and Faudree, we show that for every r ≥ 2, if ..."

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For every graph G, let ∆r (G) =max d (u):R is an r-clique of G u∈R and let ∆r (n, m) be the minimum of ∆r (G) taken over all graphs of order n and size m. Writetr (n) forthesizeofther-chromatic Turán graph of order n. Improving earlier results of Edwards and Faudree, we show that for every r ≥ 2, if m ≥ tr (n) , then ∆r (n, m) ≥ 2rm, (1) n as conjectured by Bollobás and Erdős. It is known that inequality (1) fails for m<tr (n). However, we show that for every ε>0, there is δ>0 such that if m>tr (n) − δn2 then 1

### A survey of Evasiveness: Lower Bounds on the Decision-Tree Complexity of Boolean Functions

, 1990

"... The decision tree complexity of a boolean function F of n arguments is the depth of a minimum-depth decision tree that computes F correctly on every input. A boolean function F of n arguments is c \Delta n-evasive if its decision tree complexity is c \Delta n, (c ! 1). F is completely evasive, (n-e ..."

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The decision tree complexity of a boolean function F of n arguments is the depth of a minimum-depth decision tree that computes F correctly on every input. A boolean function F of n arguments is c \Delta n-evasive if its decision tree complexity is c \Delta n, (c ! 1). F is completely evasive, (n-evasive), if all of its arguments need to be probed in the worst case. When we restrict the properties and form of the boolean functions considered, interesting results on their decision tree complexity can be derived. One such restriction is boolean functions which represent the adjacency matrix of a graph. Rivest and Vuillemin proved the Aanderaa-Rosenberg conjecture which stated that every graph property on n node graphs is c \Delta n 2 -evasive. Subsequently it was proved that all nontrivial monotone graph properties on a prime power number of nodes are completely evasive. However proving that all monotone graph properties are evasive seems a difficult task. In this paper we survey the...