10.3 Edge coloring and chromatic index

Edge coloring assigns colors to graph edges, ensuring no adjacent edges share colors. The chromatic index represents the minimum colors needed, always at least equal to the maximum graph degree. Vizing's Theorem sets an upper bound.

Edge coloring relates to matching, as each color class creates a matching. It has practical applications in scheduling and resource allocation. While NP-complete for general graphs, efficient algorithms exist for specific graph classes.

Edge Coloring Fundamentals

Edge coloring properties

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• Edge coloring assigns colors to graph edges ensuring each edge receives one color and no adjacent edges share colors
• Chromatic index $\chi'(G)$ represents minimum colors required always greater than or equal to maximum graph degree
• Proper edge coloring prevents adjacent edges from having same color while strong edge coloring prohibits same color for edges within distance 2 (path graph, cycle graph)

Chromatic index of graphs

• Chromatic index determines minimum colors for proper edge coloring with lower bound $\Delta(G)$ (maximum graph degree)
• Vizing's Theorem states $\chi'(G) \leq \Delta(G) + 1$ classifying graphs into Class 1 $\chi'(G) = \Delta(G)$ or Class 2 $\chi'(G) = \Delta(G) + 1$
• Specific graph types have known chromatic indices bipartite graphs $\chi'(G) = \Delta(G)$ complete graphs $\chi'(K_n) = n - 1$ (even n) or $\chi'(K_n) = n$ (odd n) cycles $\chi'(C_n) = 2$ (even n) or $\chi'(C_n) = 3$ (odd n)

Applications and Advanced Concepts

Edge coloring vs matching

• Matching forms set of edges without common vertices maximum matching represents largest possible set
• Edge coloring relates to matching as each color class creates a matching minimum edge coloring partitions edges into fewest matchings
• König's Theorem equates chromatic index to maximum matching size in bipartite graphs
• Edmonds' Matching Algorithm efficiently finds maximum matchings in general graphs helps determine chromatic index for some cases

Applications in scheduling and allocation

• Edge coloring solves scheduling problems timetabling (classes to time slots) sports tournament scheduling (team matches)
• Resource allocation uses edge coloring for frequency assignment (wireless networks) traffic light synchronization (transportation)
• Algorithms include greedy coloring Misra-Gries algorithm heuristic approaches (local search genetic algorithms for large-scale problems)
• Edge coloring complexity NP-complete for general graphs polynomial-time algorithms exist for specific graph classes (bipartite graphs)