We can never have more than $\frac{n}{2}$ edges of a graph $G$ of order $n$ that are colored the same. Why?
93304
How to prove Graph with $n$ vertices has Edge Chromatic Number no more than $n/2$ edges
120617
No cycle containing edges $e$ and $g$ implies there is a vertex $u$ so that every path sharing one end with $e$ and another with $g$ contains $u$
155090
A graph whose every odd cycle is a triangle is 4-colorable
79730
$G$ is a planar, connected graph. Let $n$ be number of vertices, $m$ number of edges and $f$ number of faces. Prove that dual graph $G*$ has $f(G*)=n$
52261
Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$
158856
Prove that a graph with the same number of edges and vertices contains one cycle
89452
Proof about Graph with no triangle
133270
Prove that the chromatic polynomial of a cycle graph $C_{n}$ equals $(k-1)^{n} + (k-1)(-1)^{n}$
87760
Determine the diameter of a graph of order $n \geq 3$ where $\delta (G) \geq \frac{n-1}{2}$.
124848
Why $O(n)$ has exactly two connected components?
34452
In how many ways can you choose $k$ numbers out of $\{1,2,3,\dots,n\}$ so none of them is consecutive?
18117
Maximum number of edges in a simple graph?
140444
Prove $\chi(G)\chi(\bar{G}) \geq n$ for chromatic number of graph and its complement
39753
Proving that in a simple graph with $n$ vertices, two vertices have the same degree
61340
Prove that there exists bipartite graph with this degree sequence: $(3,3,3,3,3,5,6,6,6,6,6,6,6,6)$
45193
prove or supply counter example about graph
104112
Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible?