A simple graph $G$ with $n$ vertices has at least $n+4$ edges. Prove that it has two cycles having no common edge.
131137
If ${\lvert{E(G)}\rvert}>{\lvert{V(G)}\rvert}+3$, graph $G$ has at least two disjoint cycles.
89531
Show that every graph $G$ has a bipartite subgraph with at least half of the edges of $G$
123626
Prove that even $n$ can be partitioned to $\frac n2$ edges
139657
Prove that if $\chi(G-u-v)=\chi(G)-2$ for every vertices $u, v$ ($u \ne v$) then G is complete graph.
67887
Prove that a graph on $n$ vertices with girth $g$ has at most $\frac{g}{g-2}(n-2)$ edges
4112
How to prove that a simple graph having 11 or more vertices or its complement is not planar?
68638
Show that if $n\geq 3$, the complete graph on $n$ vertices $K_n$ contains a Hamiltonian cycle.
71170
Upper Bound on the Chromatic Number of a Graph with No Two Disjoint Odd Cycles
28877
Graph with $k$ components.
48692
A problem about the graph theory
100978
If a graph with $n$ vertices and $n$ edges there must a cycle?
25382
Maximum number of edges in a bipartite graph
117685
Prove that a bipartite planar graph has a vertex of degree at most 3
145181
How can I prove the maximum number of edges?
152653
Count Eulerian cycles in $K_{n,2}$
48184
Prove a graph Containing $2k$ odd vertices contains $k$ distinct trails
145818
Edges on an even number of Hamilton cycles