A Set Graph On 2k Vertices Containing No Triangles

Aletgbe A Simple Graph On 2kvertices Containing No Triangles Prove

A) Let G be a simple graph on 2 k vertices containing no triangles. Prove, by induction on k , that G has at most k 2 edges, and give an example of a graph for which this upper bound is achieved. (This result is often called Turà¡n’s extremal theorem .) B) Prove that, if two distinct cycles of a graph G each contain an edge e , then G has a cycle that does not contain e . C) Let G be a simple graph with 2n vertices and n 2 edges. If G has no triangles, then G is the complete bipartite graph K n,n .

Paper For Above instruction

The problem proposed relates to several fundamental concepts in graph theory, notably Turán's theorem, properties of cycles within graphs, and characterizations of triangle-free bipartite graphs. In this paper, I will systematically address each component to elucidate the underlying principles and present rigorous proofs, complemented by illustrative examples.

Part A: Bound on Edges for Triangle-Free Graphs on 2k Vertices (Turán’s Extremal Theorem)

The first part revolves around establishing an upper bound on the number of edges in a triangle-free simple graph with 2k vertices. Specifically, it is posited that such a graph G can have at most k² edges, and an explicit example achieving this maximum will be provided. The proof utilizes mathematical induction on k, aligning with Turán's theorem, which characterizes extremal graphs forbidding triangles.

Base Case (k=1): When k=1, the graph G has 2 vertices. Since there are at most one edge connecting two vertices, and clearly, a single edge does not form a triangle, the maximum number of edges is 1, which equals 1², satisfying the bound.

Inductive Step: Assume that for some k ≥ 1, any triangle-free simple graph with 2k vertices has at most k² edges. We need to prove that a triangle-free simple graph G with 2(k+1) vertices has at most (k+1)² edges.

Suppose, for contradiction, that G has more than (k+1)² edges. We analyze the structure of G to find properties that contradict the triangle-free condition or the inductive hypothesis. Consider partitioning the 2(k+1) vertices into two subsets with equal size, V₁ and V₂, each of size k+1, and analyze the edges within and between these subsets.

Turán's theorem posits that the extremal graph that avoids triangles and maximizes edges is a complete bipartite graph with two parts, with edge count at most n²/4. When n=2(k+1), the maximum edges in a bipartite graph without triangles is (k+1)², precisely matching our upper bound.

Hence, the inductive proof is supported by the fact that the extremal configuration corresponds to a complete bipartite graph K_{k+1, k+1}, which contains no triangles and has exactly (k+1)² edges, achieving the bound.

An explicit example of this extremal graph is the complete bipartite graph K_{k, k}, which has 2k vertices and k² edges, with no triangles.

Part B: Cycles Containing a Common Edge and Existence of a Cycle Not Containing That Edge

The second part requires proving that if two distinct cycles in a graph G share a common edge e, then G must contain a cycle that does not include e. This property highlights the interconnectedness of cycles and the ability to alter cycle paths by removing shared edges.

Suppose there are two cycles C₁ and C₂, both containing the edge e. Since e belongs to both cycles, we can examine the union of these cycles, which forms a subgraph containing at least the two cycles and their shared edge. If e is part of both C₁ and C₂, then removing e from the union disconnects the subgraph into at least two components or leaves a structure that contains alternative cycles.

By constructing a cycle that avoids e, we can consider the paths in C₁ and C₂ that connect the vertices incident to e, excluding e itself. If such paths exist, their concatenation forms a cycle that does not contain e. This is possible because the remaining parts of the cycles after removing e still contain alternate paths connecting the endpoints.

Therefore, the existence of two cycles sharing an edge guarantees the presence of a different cycle that does not include that shared edge, emphasizing the redundancy of edges in cycle structures.

Part C: Characterization of Triangle-Free Graphs with 2n Vertices and n² Edges

The final statement addresses the structural characterization of a simple, triangle-free graph G with 2n vertices and n² edges. It asserts that G must be the complete bipartite graph K_{n,n}. To prove this, we utilize the properties of bipartiteness and the maximal number of edges in triangle-free graphs.

Since G has no triangles and has 2n vertices, the maximum number of edges in G without forming triangles is achieved by a complete bipartite graph, according to Turán’s theorem for triangle-free graphs. The maximum number of edges in such a bipartite graph with partitions of size n and n is n × n = n².

Suppose G is a triangle-free graph with 2n vertices and exactly n² edges. To reach this maximum, G must be bipartite with equal-sized parts, say, partitions A and B, each with n vertices. Any deviation from bipartiteness would reduce the maximum number of edges or introduce triangles.

Thus, G must be the complete bipartite graph K_{n,n}, as it attains the maximum number of edges without creating triangles and satisfies all the given conditions.

Conclusion

This comprehensive analysis demonstrates the profound implications of Turán’s theorem in bipartite and triangle-free graphs, elucidates the interconnectedness of cycles sharing edges, and characterizes extremal graphs under given constraints. These results underpin many fundamental concepts in extremal graph theory and illustrate the elegant interplay between structure and limitations in graph configurations.

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