Exercise · ZW-GRAPH-COLORING-008 · ★★☆ Intermediate
The Petersen graph has 10 vertices, all of degree 3, and contains 5-cycles. What is the minimum number of colors of a proper coloring?
Choose one option.
Show answer
- C. 3
Explanation
It has odd cycles, so 2 is not enough; 3 colors suffice with an explicit assignment, matching the bound \(\Delta=3\) of Brooks' theorem (it is neither complete nor an odd cycle).
Review the concept: Graph coloring
A proper coloring \(c\) with \(k\) colors of a graph \(G=(V,E)\) assigns to each vertex one of the colors \(1,\dots,k\) so that no two adjacent vertices share a color.
\[c:V\to\{1,\dots,k\},\qquad c(u)\neq c(v)\ \text{ if } \{u,v\}\in E\]
More practice problems: graph coloring
- ZW-GRAPH-COLORING-001 What is the minimum number of colors in a proper coloring of the complete graph K_5?
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- ZW-GRAPH-COLORING-004 What is the minimum number of colors needed by the cycle C_8?
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- ZW-GRAPH-COLORING-009 Let Delta be the maximum degree of a graph G. Which statement is true for every graph?