Prove the odd-cycle characterization and separate coloring from drawing aesthetics.
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Try
An even cycle can alternate two vertex colors and close consistently. An odd cycle cannot: its final edge joins equal colors. Proper coloring requires adjacent vertices differ; the colors have no numerical or visual meaning beyond that constraint.
Predict
Predict whether bipartite means equal-sized parts. It does not. A graph is bipartite when its vertices split into two sets with every edge crossing between them. K₃,₅ has parts of sizes three and five and 15 edges.
Experiment
Color a four-cycle alternately, then insert one extra vertex along an edge to create a five-cycle. Identify the closing conflict. Next draw a tree and color by even or odd distance from a root. Repeat separately for each disconnected component.
A proper three-coloring of the five-cycle uses labels 1,2,1,2,3 around A,B,C,D,E. These numeric labels represent colors, so the explanation does not depend on actual display color.
Understand
Worked characterization: a bipartite graph has no odd cycle because every edge switches parts, so a return needs an even number of switches. Conversely, root each connected component and group vertices by parity of shortest-path distance. If an edge joined two vertices of the same parity, their tree paths to the last common ancestor plus that edge would give an odd cycle. Thus a graph with no odd cycle is bipartite.
A general graph of maximum degree Δ can be greedily colored with Δ+1 colors. At each step at most Δ neighbor colors are forbidden, leaving one available. This proves an upper bound, not optimality.
Invalid claim: an even number of vertices guarantees bipartiteness. A triangle with one isolated vertex has four vertices but still contains an odd cycle.
Apply
Give both a coloring and a lower-bound obstruction when claiming an exact chromatic number. A displayed coloring proves only that many colors suffice. A triangle proves at least three are necessary, and an odd cycle rules out two.
Continue
Depth: edge coloring, matching and Hall’s theorem ask different questions and are part of advanced combinatorics. A bipartition can guide matching models, but its existence alone does not guarantee every vertex can be matched.
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Practice and reasoning
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How many edges does K₃,₅ have?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
What is the chromatic number of C₅?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A finite graph is bipartite exactly when it has:
A proper vertex coloring requires:
Prove a bipartite graph cannot contain an odd cycle.
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Prove that a finite simple graph of maximum degree Δ can be properly colored using at most Δ+1 colors.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
