Strong coloring 2-regular graphs: Cycle restrictions and partial colorings
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Let H be a graph with Δ(H) ≤ 2, and let G be obtained from H by gluing in vertex-disjoint copies of K_4. We prove that if H contains at most one odd cycle of length exceeding 3, or if H contains at most 3 triangles, then χ(G) ≤ 4. This proves the Strong Coloring Conjecture for such graphs H. For graphs H with Δ=2 that are not covered by our theorem, we prove an approximation result towards the conjecture.
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