**Eulerian Graphs and Blocks** Let \( G \) be a connected graph. Prove that \( G \) is Eulerian if and only if each of the blocks of \( G \) is Eulerian. In this context, an *Eulerian graph* is a graph containing a closed trail that visits every edge of the graph exactly once. A *block* in a graph is a maximal connected subgraph that has no cut-vertex, which means it is connected and remains connected if any single vertex is removed. To prove this statement, one needs to show both directions: - If \( G \) is Eulerian, then each block of \( G \) is Eulerian. - If each block of \( G \) is Eulerian, then \( G \) is Eulerian. This involves examining the properties of Eulerian graphs, degree conditions, and the structure of blocks within the graph \( G \).

Prove that two cycles (a1,a2,...ak), (b1,b2,...bm) are disjoint cycles.

Let G be a graph. Prove that if G has a closed walk of odd length, that the vertices and edges of the walk contain a cycle.

(Discrete Math-Graph Theory) Let G=(V,E₁∪E₂∪E₃) be a simple graph such that G₁=(V,E₁) is planar, G₂=(V,E₂) is a forest, and G₃=(V,E₃) is a matching. Prove G is 9-colourable, i.e. its chromatic number satisfies χ(G)≤9.

Can someone provide the proof for this? TIA

Consider an undirected graph G = (V, E). An independent set is a subset I CV such that for any vertices i, j e I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let a(G) denote the size of the largest maximal independent set in G.

The symmetric difference graph of two graphs G. (V. E.) and G₂ (V₁ E₂) on the same vertex Set is defined as G₁ AG₂:= (V, E, DE₂). E₁ DE ₂ = (E₁ \ E₂) U (E₂\E.) : E₁ E₂ = €₁ 0 € ₂ If G₁ and 6₂ are euleran, show that every Vertex in G, D G₂ has even degree FACT: Each vertex of a evlenan graph has even degree. SO: Show G₁ D G₂ is eulerian.