B.5.4:5.2 - Recognize a conflict relation before colouring a graph

A laboratory must place three tests in two simultaneous-run slots. Its operating conditions say that A and B need the same exclusive fixture throughout a run; B and C need the same exclusive power unit; A and C can run together. There are no other constraints in this constructed case.

Represent tests by vertices and a pairwise inability to share a slot by an edge. The observations supply edges AB and BC. The construction in B.5:5.1 produces slots {A,C} and {B}; reading the result back means that neither exclusive resource is double-booked.

Now change the conditions: A and C also require the same observer throughout a run. This adds edge AC. The triangle needs three slots under these conditions; two-colouring no longer supplies an arrangement.

Consider instead three tests with independent fixtures whose shared resource is a 5 A supply. Each draws 2 A. Any pair can run together, but all three exceed the supply’s capacity. A graph with no pairwise conflict edges loses that group constraint. Retain the currents and the slot-wise capacity inequality when constructing the arrangement.