Not complementary connected and not CIS -graphs form weakly monotone families

A d -graph G = ( V ; E 1 , … , E d ) is a complete graph whose edges are arbitrarily partitioned into d subsets (colored with d colors); G is a Gallai d -graph if it contains no 3-colored triangle Δ ; furthermore, G is a CIS d -graph if ⋂ i = 1 d S i ≠ 0̸ for every set-family S = { S i | i ∈ [ d ] } , where S i ⊆ V is a maximal independent set of G i = ( V , E i ) , the i th chromatic component of G , for all i ∈ [ d ] = { 1 , … , d } . A conjecture suggested in 1978 by the third author says that every CIS d -graph is a Gallai d -graph. In this article, we obtain a partial result. Let Π be the 2-colored d -graph on four vertices whose two non-empty chromatic components are isomorphic to P 4 . It is easily seen that Π and Δ are not CIS d -graphs but become CIS after eliminating any vertex. We prove that no other d -graph has this property, that is, every non-CIS d -graph G distinct from Π and Δ contains a vertex v ∈ V such that the sub- d -graph G [ V ∖ { v } ] is still non-CIS. This result easily follows if the above Δ -conjecture is true, yet, we prove it independently. raph G = ( V ; E 1 , … , E d ) is complementary connected (CC) if the complement G ¯ i = ( V , E ¯ i ) = ( V , ⋃ j ∈ [ d ] ∖ { i } E j ) to its i th chromatic component is connected for every i ∈ [ d ] . It is known that every CC d -graph G , distinct from Π , Δ , and a single vertex, contains a vertex v ∈ V such that the reduced sub- d -graph G [ V ∖ { v } ] is still CC. not difficult to show that every non-CC d -graph with at least two vertices contains a vertex v ∈ V such that the sub- d -graph G [ V ∖ { v } ] is not CC.