Density of 5/2-critical graphs


A graph G is 5/2-critical if G has no circular 5/2-coloring (or equivalently, homomorphism to C 5), but every proper subgraph of G has one. We prove that every 5/2-critical graph on n ≥ 4 vertices has at least

$$\frac{{5n - 2}}{4}$$

edges, and list all 5/2-critical graphs achieving this bound. This implies that every planar or projective-planar graph of girth at least 10 is 5/2-colorable.

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Corresponding author

Correspondence to Zdeněk Dvořák.

Additional information

Supported by project GA14-19503S (Graph coloring and structure) of Czech Science Foundation.

Partially supported by NSERC under Discovery Grant No. 2014-06162, the Ontario Early Researcher Awards program and the Canada Research Chairs program.

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Dvořák, Z., Postle, L. Density of 5/2-critical graphs. Combinatorica 37, 863–886 (2017).

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Mathematics Subject Classification (2000)

  • 05C15