Planar character-graphs.

For a finite group G, let R(G) be the solvable radical of G. The character-graph Δ (G) of G is a graph whose vertices are the primes which divide the degrees of some irreducible complex characters of G and two distinct primes p and q are joined by an edge if the product pq divides some character degree of G. In this paper we prove that, if Δ (G) has no subgraph isomorphic to K 3 , 3 and it's complement is non-bipartite, then G R (G) is an almost simple group with socle isomorphic to PSL 2 (q) where q ⩾ 5 is a prime power. Also we study the structure of all planar graphs that occur as the character-graph Δ (G) of a finite group G. [ABSTRACT FROM AUTHOR]