Exercise · ZW-GRAPH-COLORING-009 · ★★★ Advanced
Let \(\Delta\) be the maximum degree of a graph \(G\). Which statement is true for every graph?
Choose one option.
Show answer
- A. \(G\) can be colored with \(\Delta+1\) colors
Explanation
A greedy coloring always finds a free color among \(\Delta+1\), since a vertex has at most \(\Delta\) neighbors. \(K_4\) refutes “it can be colored with \(\Delta\) colors” (it needs 4 with \(\Delta=3\)) and a path refutes “it needs at least \(\Delta+1\) colors” and “it needs as many colors as vertices”.
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?
- ZW-GRAPH-COLORING-002 What is the minimum number of colors needed by a graph that has at least one edge?
- ZW-GRAPH-COLORING-003 What is the minimum number of colors needed by the cycle C_7 (7 vertices)?
- ZW-GRAPH-COLORING-004 What is the minimum number of colors needed by the cycle C_8?
- ZW-GRAPH-COLORING-005 The graph has vertices 1,2,3,4 and edges \1,2\,\2,3\,\1,3\,\3,4\. Which is a proper coloring?
- ZW-GRAPH-COLORING-006 The wheel W_5 is a cycle C_5 plus a central vertex joined to all 5 cycle vertices. How many colors…
- ZW-GRAPH-COLORING-007 In how many ways can a path on 4 vertices be colored with 3 available colors so that adjacent verti…
- ZW-GRAPH-COLORING-008 The Petersen graph has 10 vertices, all of degree 3, and contains 5-cycles. What is the minimum num…