(c) Drawwith explanation.
The graph $$ C_4 \times S_4 $$ is the **Cartesian product** of the cycle graph $$ C_4 $$ (a cycle with 4 vertices) and the star graph $$ S_4 $$ (a central vertex connected to 4 outer vertices). *** ## Step 1: Understand the Components - $$C_4$$: A cycle with 4 vertices connected in a closed loop: Vertices: $$ \{c_1, c_2, c_3, c_4\} $$ Edges: $$ (c_1, _______ __________ __________ _________ ___ __________ _______ ______ _____ _________.
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(d) Provide an example of a 3-regular planar graph with 8-vertices. Is this graph a maximal planar graph? Why?
(c) State and prove Hall’s Theorem
State whether the following statements are true or false. Justify your answers with a short proof or a counterexamp
i) There exists an 8-vertex graph with three vertices of degree 3, four vertices of degree 2 and one vertex of degree 1
ii) The neighbour of every leaf is a cut-vertex.
iii) Every line graph of a bipartite graph is 2-colourable
iv) is a graphic sequence then so is
v)
vi) A Hamiltonian graph has no cut-vertices.
vii) The Petersen graph is 3-critical.
viii) An n-vertex star has no perfect matching for n ≥ 3.
ix) The crossing number of K3,3 is 2.
x) If f and g are two flows on a network N, then max is also a flow.
(d) Find the matching number of the line graph of the graph given in part(a).
(a) Verify Euler’s formula for the following plane graph.
(a) What is the maximum possible flow that can pass through the following network N? Define such a flow
(c) Prove or disprove: If G is a graph with χ(G) denoting its chromatic number, then