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Definition 7.1.3

Definition 7.1.3. k-edge-coloring of G: A labeling f: E(G) -> S, where |S| =k. The labels are colors; the edges of one color form a color class. proper k-edge-coloring: A k-edge-coloring such that incident edges have different labels.

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Definition 7.1.3

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  1. Definition 7.1.3 • k-edge-coloring of G: A labeling f: E(G) -> S, where |S| =k. The labels are colors; the edges of one color form a color class. • proper k-edge-coloring: A k-edge-coloring such that incident edges have different labels. • k-edge-colorable graph: A graph having a proper k-edge-coloring. • edge-chromatic number, x’(G), of G: The least k such that G is k-edge-colorable.

  2. Theorem 7.1.10 • If G is a simple graph, then x’(G) ≤Δ(G)+1. Proof: 1.Let f be a proper Δ(G)+1-edge-coloring of a subgraph G’ of G. 2. If G’≠ G, then some edge uv is uncolored by f. 3. It suffices to show we can extend the coloring to include uv after possibly recoloring some edges (an augmentation).

  3. a0 u a1 v1 v0= v a1 Theorem 7.1.10 4.Every vertex has some color not appearing on its incident edges because the number of colors exceeds Δ(G). 5. Let a0 be a color missing at u. 6. Let a1 be a color missing at v0(=v). We may assume that a1 appears at u on some edge uv1; otherwise, we would use a1 on uv0.

  4. a0 a3 u v2 a2 a1 a2 a1 v1 v0= v Theorem 7.1.10 7. Let a2 be a color missing at v1. We may assume that a2 appears at u on some edge uv2; otherwise, we would replace color a1 with a2 on uv1 and then use a1 on uv0 to augment the coloring.

  5. ak+1 a1 Theorem 7.1.10 8. Having selected uvi-1 with color ai-1, let ai be a color missing at vi-1. If ai is missing at u, then we use ai on uvi-1 and shift color aj from uvj to uvj-1 for 1≤ j ≤i-1 to complete the augmentation. We call this downshifting from vi-1. If ai appears at u (on some edges uvi), then the process continues. ai ai-1 u vi-1 ak vk v0= v

  6. ak al u a0 vl ak ak+1 ak ak-1 a1 vk vk-1 v0= v Theorem 7.1.10 9. Since we have only Δ(G)+1 colors to choose from, the list of selected colors eventually repeats (or we complete the augmentation by downshifting). 10. Let l be the smallest index such that a color missing at vl is in the list a1,…,al; let this color be ak. 11. The color ak missing at vl is also missing at vk-1 and appears on uvk. If a0 does not appear at vl, then we downshift from vl and use color a0 on uvl to complete the augmentation. Hence we may assume that a0 appears at vl.

  7. Theorem 7.1.10 12. Let P be the maximal alternating path of edges colored a0 and ak that begins at vl along color a0. There is only one such path because each vertex has at most one incident edge in each color (we ignore edges not yet colored).

  8. Theorem 7.1.10 13. If P reaches vk, then it arrives at vk along an edge with color a0, follows vku in color ak, and stops at u, which lacks color a0. In this case, we downshift from vk and switch colors on P.

  9. Theorem 7.1.10 14. If P reaches vk-1, then it reaches at vk-1 on color a0, and stops there, because ak does not appear at vk-1. In this case, we downshift from vk-1, give color a0 to uvk-1, and switch colors on P.

  10. Theorem 7.1.10 15. If P does not reach vk or vk-1, then it ends at some vertex outside {u, vl, vk, vk-1}. In this case, we downshift from vl, give color a0 to uvl, and switch colors on P.

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