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

A d-graph G = (V ; E1, . . . , Ed) 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 three-colored triangle ∆; furthermore, G is a CIS d-graph if ⋂d i=1 Si 6= ∅ for every set-family S = {Si | i ∈ [d]}, where Si ⊆ V is a maximal independent set of Gi = (V,Ei), the ith 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 paper we obtain a partial result. Let Π be the two-colored dgraph on four vertices whose two non-empty chromatic components are isomorphic to P4. It is easily seen that Π and ∆ 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 ∆ 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 ∆-conjecture is true, yet, we prove it independently. A d-graph G = (V ; E1, . . . , Ed) is complementary connected (CC) if the complement Gi = (V,Ei) = (V, ⋃ j∈[d]\{i}Ej) to its ith chromatic component is connected for every i ∈ [d]. It is known that every CC d-graph G, distinct from Π, ∆, and a single vertex, contains a vertex v ∈ V such that the reduced sub-d-graph G[V \ {v}] is still CC. It is not difficult to show that every non-CC d-graph with contains a vertex v ∈ V such that the sub-d-graph G[V \ {v}] is not CC.