Let G=(V,E) be a simple graph with n vertices and m edges, P(G,k) the chromatic polynomial of G, P(G,L) number L-colorings G for any k-assignment L. In this article, we show that when k≥m−1≥3, P(G,L)−P(G,k) is bounded below by ((k−m+1)kn−3+(k−m+3)c3kn−5)∑uv∈E|L(u)∖L(v)|, where c≥(m−1)(m−3)8, in particular, if K3-free, then c≥(m−22)+2m−3. Consequently, P(G,L)≥P(G,k) whenever k≥m−1.

We study a variant of the inverse problem of Galois theory and Abhyankar’s conjecture. If p is an odd rational prime and G is a finite p-group generated by s elements, s minimal, does there exist a normal extension L/Q such that Gal (L/Q) ∼= G with at most s rational primes that ramify in L? Given a nilpotent group of odd order G with s generators, we obtain a Galois extension L/Q with precisel...

A crossed product is the functional analysts’ version of a skew group ring. Thus, if α : G→ Aut(A) is an action of a locally compact group G on a Banach algebra A, then a crossed product Banach algebra encodes the action of G on A, and its representation theory is related to pairs (u, π) consisting of a representation u of G and a representation π of A on the same Banach space such that ugπ(a)u...