Prove that in any graph G with n vertices, n ≤ α(G)χ(G)

Q: Prove or disprove: There exist two connected graphs G and H both of order at least 3 and neither of…

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Q: 5. Let G = (V, E) be a finite, connected graph. Show that if no vertex of G has degree less than 2,…

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Q: Let G be a simple graph with 15 vertices and 4 connected components. Prove that G has at least one…

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Q: Prove that if G is a graph with no isolated vertices, then α'(G) +β'(G) = n(G)

A: Here α'(G) is largest size of matching in G. β'(G) is the smallest size of an edge cover. We have to…

Q: 5. Prove that every graph of order 6 with chromatic number 3 has at most 12 eges.

A: Solution :- As per guidelines.i solve 5th number.

Q: prove that the maximum girth of a generalized Coxeter graph is 12, no matter what its parameters are

A: Among the generalized Petersen graphs, having n prism G p,1 on the Durer graph G 6.2 the…

Q: Let G be the (3,4)-complete bipartite graph. (a) List all non-isomorphic subgraphs with 5 vertices…

A: Part(a) Solution:Let be the (3,4) complete bipartite graph K3,4      These are the 11…

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

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Q: Let G be a simple graph on 9 vertices, and assume we know that the sum of all degrees in G is at…

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Q: Prove that having n vertices, where n is a positive integer, is an invariant for graph isomorphism.

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Q: 7. Suppose a graph G = (V, E) has n vertices and m edges (i.e., |V| = n and |E| = m). Further…

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Q: Prove on the number of vertices that every graph G with n edges has a bipartite subgraph with at…

A: We will first assume that  P(n) : every graph G with n edges has a bipartite subgraph with atleast…

Q: If G is a colored graph in which each vertex is assigned a color. The chromatic number of a graph G,…

A: The chromatic number of a graph χG is the minimum number of colors needed in a graph to color it so…

Q: 4.1.3. (-) Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then…

A: Given: Let G be an n-vertex simple graph other than Kn. Prove that if G is not k-connected, then G…

Q: 18. Let G be a graph with n vertices and exactly n-1 edges . Prove that G has either a vertex of…

A: Now, we need to prove that if G is a graph with n vertices and (n-1) edges then G has either a…

Q: Draw all non-isomorphic graphs of order n with 1<n<4.

A: As per the question we have to draw all the non-isomorphic graphs of order 1, 2, 3, 4 But as there…

Q: Let G be an undirected graph with n vertices. Let A(G) be the maximum degree of any vertex in G,…

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Q: b) Is the following graph isomorphic to Q4? If yes, give an isomorphism from Q4 to this graph.…

A: 2b ) Given graph is Here number of vertices in the given graph is 12 and number of vertices in Q4…

Q: G is a graph on n 21 vertices such that both G and its complement have an Euler trail. (Recall that…

A: To do: Draw a simple graph G such that "G is a graph with n≥1 vertices such that both G and its…

Q: Let G be a connected simple graph in which all vertices have degree 4. Prove that it is possible to…

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Q: Q10. Using Corollary 3.1.24 of the book, to prove that a graph G is bipartite if and only if a(H) =…

A: Given: α(H)=β'(H) To find: Isolated vertices

Q: Prove that a bipartite graph has a unique bipartition iff it is connected.

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Q: Let G be a connected graph with 3 or more vertices and such that x(G) = 3. Consider a proper…

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Q: 15. Let G be a graph on n vertices (with no loops or multiple edges). Prove that if all vertices in…

A: 1. A graph G is an ordered pair (V(G), E(G)) consisting of a non-empty set V(G) of vertices and a…

Q: (b) Show that G may not necessarily have three cycle subgraphs by giving an example of a graph G,…

A: Consider the equation

Q: 5.1.12. (-) Prove or disprove: Every k-chromatic graph G has a proper k-coloring in which some color…

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Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. a)Prove that…

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Q: 11. > Prove: If G is a simple connected graph where the average degree of the vertices is exactly 2,…

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Q: 6. A vertex v of G is essential if v is covered by every maximum matching in G. That is, a'(G - v) =…

A: Given: Here the given vertex v of G is essential if v is covered by every maximum matching in G.…

Q: Show that every simple graph on n vertices subgraph of K.

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Q: 5. Let G be a connected graph without loops and of order at least 11. Prove that G or its complement…

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Q: Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to…

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Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. Prove that if…

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Q: 3. If G is a colored graph in which each vertex is assigned a color. The chromatic number of a graph…

A: If G is colored graph assignment is the chromatic number the graph assignment XG is the least number…

Q: Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced…

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Q: Prove: “The complement of the graph G is an Eulerian graph whenever G is also  an Eulerian graph of…

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Q: Let G be a 2-connected graph. Prove that if G has no even cycles, then G is isomorphic to an odd…

A: k-Connected graphs: A graph is said to be k-connected if it has more than k vertices and remains…

Q: A (general) bipartite graph G is a simple graph whose vertex set can be partitioned into two…

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Q: Let G be an odd cycle. Prove that any proper 3-coloring of G using the colors red, green, and blue…

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Q: 5.1.21. Suppose that every edge of a graph G appears in at most one cycle. Prove that every block of…

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Q: =. Find the smallest 3-regular graph G such that (G) 1.

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Q: Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and…

A: We are given a G graph G with v number of vertices and e number of edges in the graph. we are also…

Q: For any connected graph G of order 54, if x is a vertex of G, then the number of components of G-x…

A: Given: To explain the given statement as follows,

Q: Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for…

A: If we are confident with our definition we can easily attempt like these questions, G is connected…

Q: 9. A bipartite graph G is a simple graph whose vertex set can be partitioned into two disjoint…

A: Since you have asked a question with multiple subparts so as per guidelines we will solve the first…