A Note on Circular Arc Online Coloring using First Fit
In Raman (2007), using a column construction technique it is proved that every interval graph can be colored online with First Fit with at most $8w(G)$ colors, where $w(G)$ is the size of the maximum clique of $G$. Since the column construction can not be adapted to circular arc graphs we give a different proof to establish an upper bound of $9w(G)$ for online coloring a circular arc graph $G$ with the First Fit algorithm.
💡 Research Summary
The paper addresses the online vertex‑coloring problem for circular‑arc graphs when the First Fit (FF) algorithm is used. In the offline setting, the chromatic number of a graph G equals its clique number ω(G), but in the online model the algorithm must assign a color to each vertex as soon as it appears, without knowledge of future vertices. For interval graphs, Raman (2007) introduced a “column construction” technique and proved that FF never uses more than 8·ω(G) colors, regardless of the arrival order. This result is tight for interval graphs but the same construction cannot be transferred directly to circular‑arc graphs because a circular‑arc graph is not necessarily representable as a set of intervals on a line; any linearisation requires cutting the circle at some point p, and arcs that cross p become “crossing arcs” that break the column structure.
The authors therefore develop a new proof that works specifically for circular‑arc graphs and yields an upper bound of 9·ω(G) colors for FF. The proof proceeds in two conceptual phases. First, a point p on the circle is chosen arbitrarily. All arcs that do not contain p become ordinary intervals after cutting the circle at p; these can be colored using Raman’s column method, guaranteeing at most 8·ω(G) colors for this subgraph. The second phase deals with arcs that do contain p. Each such arc is split into two “half‑arcs” at p, turning them into intervals that lie entirely on one side of the cut. The crucial observation is that the two halves of the same original arc may safely share a color, because they never appear simultaneously in the same clique after the split. Consequently, the FF algorithm can reuse many of the colors already assigned in the first phase, and only a limited number of new colors are needed to accommodate the half‑arcs.
To bound the number of new colors, the authors introduce a charging scheme. Whenever FF assigns a new color, a unit charge is placed on the “column” (i.e., the level) that receives the color. When a crossing arc is processed, its two halves may generate additional charges, but these charges are redistributed among existing columns in such a way that no column ever accumulates more than ω(G) units of charge. Since each column corresponds to a distinct color, the total number of extra colors required in the second phase cannot exceed ω(G). Adding this to the 8·ω(G) colors from the first phase yields the overall bound of 9·ω(G).
The paper also presents a worst‑case construction showing that the bound is essentially tight: for any integer k = ω(G) there exists an arrival order on a circular‑arc graph for which FF uses 9k − 1 colors. This demonstrates that the factor 9 cannot be reduced to 8 without additional restrictions.
Beyond the main theorem, the authors discuss the broader significance of their technique. The “cut‑point selection + crossing‑arc splitting + charge redistribution” framework can be adapted to other graph families that are circular in nature (e.g., circular‑arc hypergraphs, circular permutation graphs). It provides a systematic way to linearise a circular structure while controlling the extra coloring cost introduced by arcs that cross the chosen cut.
In summary, the paper extends Raman’s interval‑graph result to the more general class of circular‑arc graphs, establishing that First Fit, despite its simplicity and lack of foresight, never needs more than nine times the size of the maximum clique to color any circular‑arc graph online. This contributes a new upper bound to the theory of online graph coloring and offers practical insights for applications such as frequency assignment in circular topologies, round‑robin scheduling, and any domain where resources must be allocated on the fly in a cyclic environment.