Question

Let ? be a connected graph with at least one edge.

**(a)** Prove that each vertex of ? is saturated by
some maximum matching in ?.

**(b)** Prove or disprove the following: Every edge
of ? is in some maximum matching of ?.

Answer #1

(a) Let L be a minimum edge-cut in a connected graph G with at
least two vertices. Prove that G − L has exactly two
components.
(b) Let G an eulerian graph. Prove that λ(G) is even.

Prove or disapprove each of the following:
(a) Every disconnected graph has an isolated vertex.
(b) A graph is connected if and only if some vertex is connected
to all other vertices.
(c) If G is a simple, connected, Eulerian graph, with edges e, f
that are incident to a common vertex, then G has an Eulerian
circuit in which e and f appear consequently.

Graph Theory
Let v be a vertex of a non trivial graph G. prove that if G is
connected, then v has a neighbor in every component of G-v.

let G be a connected graph such that the graph formed by
removing vertex x from G is disconnected for all but exactly 2
vertices of G. Prove that G must be a path.

Prove that if k is odd and G is a k-regular (k −
1)-edge-connected graph, then G has a perfect matching.

Let G be a graph where every vertex has odd degree, and G has a
perfect matching. Prove that if M is a perfect matching of G, then
every bridge of G is in M.
The Proof for this question already on Chegg is wrong

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

Let G be a simple graph having at least one edge, and let L(G)
be its line graph. (a) Show that χ0(G) = χ(L(G)). (b) Assume that
the highest vertex degree in G is 3. Using the above, show Vizing’s
Theorem for G. You may use any theorem from class involving the
chromatic number, but no theorem involving the chromatic index

Let G be an n-vertex graph with n ≥ 2 and δ(G) ≥ (n-1)/2. Prove
that G is connected and that the diameter of G is at most two.

a.
An edge in an undirected connected graph is a bridge if removing
it disconnects the graph. Prove that every connected graph all of
whose vertices have even degrees contains no bridges.
b.Let r,s,u be binary relations in U. Verify the following
property: if both relations r and s are transitive then the
intersection of r and s is transitive too.

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