Extremal Bicyclic 3-Chromatic Graphs

Abstract

A partial order relation in the set $$\mathcal {G}(n,k)$$G(n,k) of graphs of order $$n$$n and chromatic number $$k$$k can be defined as follows: Let $$G$$G and $$H$$H be two graphs in $$\mathcal {G}(n,k)$$G(n,k). $$G$$G is said to be less than $$H$$H if $$c_i(G)\le c_i(H)$$ci(G)≤ci(H) holds for every $$i$$i, $$k\le i\le n$$k≤i≤n and at least one inequality is strict, where $$c_i(G)$$ci(G) denotes the number of $$i$$i-color partitions of $$G$$G. These numbers are the coefficients of the chromatic polynomial in factorial form. In (J Graph Theory 43:210---222, 2003) the first $$\lceil n/2\rceil $$?n/2? levels of the diagram of the partially ordered set of connected 3-chromatic graphs of order $$n$$n were described. In this paper the previous work is continued and a description of the $$(\lceil n/2\rceil +1)$$(?n/2?+1)-st level is given; it contains $$n/2+1$$n/2+1 bicyclic graphs for even $$n$$n and $$(n-1)/2$$(n-1)/2 bicyclic graphs for odd $$n$$n. Some consequences concerning ordering chromatic polynomials of these graphs are deduced.