Consider a graph of nodes and non-directed edges and suppose that we are required to give each node a label, with the constraint that the labels on two nodes directly connected by an edge must be different. The labels on two nodes not directly connected may be the same or different. Is there any way of determining the minimum number of distinct labels required for the labelling? We do not require that the graph be completely connected, so some nodes may not be connected to any other.
2026-05-14 05:00:33.1778734833
Minimal labelling of the nodes of a graph where connected nodes must have the different labels.
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What you are referring to is called graph coloring, and is a whole field of research in the field of graph theory. See here for an introduction.
Perhaps the most known result on the topic is the 4 color theorem which states that planar graphs can always be colored by 4 colors (or, of course, fewer in some cases).