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For a finite group G, let Δ(G) denote the character graph built on the set of degrees of the irreducible complex characters of G. In this article, we determine the structure of all finite groups G with K4-free character graph Δ(G) having seven vertices. We also obtain a classification of all K4-free graphs with seven vertices which can occur as character graphs of some finite groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the character graph Δ(G) of a finite solvable group G. We prove that sum of the chromatic number of Δ(G) and the matching number of complement graph of Δ(G) is equal to the order of Δ(G). Also, we prove that when Δ(G) is not a block, the chromatic number of Δ(G) is equal to the clique number of Δ(G). [ABSTRACT FROM AUTHOR]