Prove that a graph is two-colorable (bipartite) if and only if it contains no odd-length cycle

Q: If G1 and G2 are isomorphic, must they have the same number of vertices? If G1 and G2 have the same…

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Q: Find all r-regular graphs of order n for a convenient selection of r, n.

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Q: Prove that for every coloring (n colors) of a complete graph on 2n+1 vertices there's a…

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Q: Prove that the following two graphs are isomorphic using properties of incidence and adjacency…

A: The star graph with the labels is as follows: The adjacency matrix of the star graph is as follows:…

Q: Prove this corollary... The connected graph G contains an Eulerian trail if and only if there are at…

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Q: Prove that if every edge of a graph G lies on an odd number of cycles, then G is Eulerian.

A: We prove that if every edge of a graph G lies on an odd number of cycles, then G is Eulerian as…

Q: Find a graph G with seven vertices such that G and its complement both have an Euler cycle.

A: Step:- 1 Euler Cycle:- is a cycle in which an Eulerian trail starts and ends on the same vertex.…

Q: 1. Suppose G is a connected graph. Show that G- e is disconnected iff the edge e does not lie on a…

A: Let G be a connected graph.  Let e be an edge in G such that G' = G – e is disconnected.  Let, if…

Q: 4. Prove the following theorem: If a graph has more than two vertices of odd degree, then it does…

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Q: Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane…

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Q: (d) Show that the pair of graphs are not isomorphic by showing that there is a property that is…

A: (d)  To prove that the pair of graphs are not isomorphic by showing that there is        a property…

Q: Prove that a cycle with even number of vertexes is bipartite graph.

A: A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U…

Q: Euler cycle.

A: AnswerSuppose we have some simple graph G, with odd n>, 1 nodes, and that this graphhas an Euler…

Q: The most famous theorem in graph theorem is the following: Theorem 1. The vertices of a planar graph…

A: No, it is not true that all graphs with χG<4 are planar. it is known that Peterson graph is not…

Q: Show that there is a graph G with exactly 5 nodes where both G and its complement, have chromatic…

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Q: (7) Verify whether it is possible to have a connected Eulerian graph with 3 vertices and 6 edges,…

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Q: Draw a cycle with 3 vertice

A: a cycle in graph theory is a graph where the start and end of the vertex is same. It will form a…

Q: e say that a graph is k-regular if each of its vertices is the endpoint of exactly k edges. Consider…

A: A k-regular bipartite graph G=X⊔Y, E, where 1≤k≤n. A graph is said to be k-regular if each of its…

Q: Prove the statement: If G is a bipartite graph, then it does not contain cycles of odd length.  Is…

A: if G is bipartite with vertex sets V1 and V2 every step along a walk takes you either from V1 to V2…

Q: Draw an Eulerian graph which has 5 vertices having one vertex with degree 4 and other vertices with…

A: Given: To draw the Eulerian graph which has 5 vertices having one vertex with a degree 4 and the…

Q: A K- regular graph is a graph where every vertex has degree K

A: Consider the given information. A graph is hamiltonian if it visits every vertex exactly once except…

Q: A cycle in a simple graph is a closed walk of length greater than m. What is the value of m?

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Q: How can I prove "A graph is k-colourable if and only if it is k-partite.”?

A: As per the question we have to prove that a graph is k-colourable if and only if it is k-partite. To…

Q: Prove that every cycle in the bipartite graph have even number of edges.

A: A bipartite graph is a graph which contains two vertex sets. Let G be a bipartite graph and let  X…

Q: 3. (Exercise 3.4) Prove Lemma 3.3: "Let G be a 2-connected graph. If e and f are parallel edges in…

A: Let G be a 2-connected graph. e and f are parallel edges in G. Let H= G\e We have to prove that if…

Q: subgraph of a nonplanar graph is nonplanar

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Q: The graph of wheel denoted by W, is obtained when an additional vertex is added to cycle Cn, for n >…

A: We need to match each statement (a) - (d) to and a graph (1)-(5) such that the chosen graph…

Q: Prove that every hamiltonian bipartite graph is an equally bipartite

A: Let G be a Hamiltonian bipartite graph with bipartition (X, Y). We want to show that |X| = |Y|.

Q: An outerplanar graph is a graph that admits a planar embedding with all vertices on the boundary of…

A: Given that an outerplanar graph is a graph that admits a planar embedding with all vertices on the…

Q: Suppose that G is 3-connected. Prove that G must have an even cycle as a sub graph.

A: A graph is said to have an even cycle, if the number of edges in the cycle is of even length A…

Q: Let G be a graph of order n ≥ 3. Prove that G is 2-connected if and only if for any vertex v and…

A: Proof:Sufficiency:Suppose that G is 2-connected and for any vertex v and edge e of G, v and e lie on…

Q: 2.5 Prove that if a graph has exactly two vertices of odd degree, then there is a path from one of…

A: In graph theory, a graph consists of vertices and edges.The degree of a vertex is the number of…

Q: Draw a simple graph G with 7 vertices and δ(G) ≥ 3 that has no Eulerian cycle, but has an Eulerian…

A: In the given question we have to draw a graph G with 7 vertices and δ(G)≥3has no eulerian cycle but…

Q: Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a…

A: First, we need to understand the problem. We are given two paths P1 and P2 of maximum length in a…

Q: Prove that cycles are bipartite for an even number of vertexes. But they are not bipartite for the…

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Q: Prove that the cartesian product of two eulerian graphs is an eulerian graph. hence 2n-cube Q2n is…

A: We are to prove that the cartesian product of two eulerian graphs is an eulerian graph, hence…

Q: Prove every bipartite graph with an odd number of vertices does not have a Hamilton cycle/circuit

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Q: Prove that if a graph on n vertices is connected, and has n-1 edges, it must be acyclic.

A: A graph G is said to be a connected graph, if there is a path between any two vertices in that…

Q: Complete a path straight at vertex a that ends at vertex c.

A: A complete path is defined when all the vertices are passed through and none of the smaller paths…

Q: Show that if G is a connected graph and e is an edge in G that is part of a cycle, then the graph G'…

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Q: (a) (b) (c)

A: a) according to question,to prove that any graph with ɛ > v contains a cycle, we can use the…