Prove that if G is a graph with $\chi$(G-v-w)=$\chi$(G)-2 for every pair of vertices v and w in G, then G is complete.
139657
Prove that if $\chi(G-u-v)=\chi(G)-2$ for every vertices $u, v$ ($u \ne v$) then G is complete graph.
37984
Prove a planar graph with all regions having even degree can be colored with 2 colors
117992
Help with graph induction proof
49879
A bipartite graph with a perfect matching has a vertex with each edge contained in a perfect matching
90747
Upper bound on $\chi(G)$ for a triangle-free graph
31854
Proof - raising adjacency matrix to $n$-th power gives $n$-length walks between two vertices
140444
Prove $\chi(G)\chi(\bar{G}) \geq n$ for chromatic number of graph and its complement
62077
Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
110422
Prove that a graph $G$ is a forest if and only if every induced subgraph of $G$ contain a vertex of degree at most $1$
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 $f(G*)=n$
31626
proving that a graph is hamiltonian (Ore's theorem)
113022
Is the graph $G_f=\{(x,f(x)) \in X \times Y\ : x \in X \}$ a closed subset of $X \times Y$?
89452
Proof about Graph with no triangle
42367
Prove that simple graph $G$ is orientable
7626
Proving that every connected graph has a vertex whose removal will not disconnect the graph.
133270
Prove that the chromatic polynomial of a cycle graph $C_{n}$ equals $(k-1)^{n} + (k-1)(-1)^{n}$
33316
Show that if $G$ is simple a graph with $n$ vertices and 􏰈the number of edges $m>\binom{n-1}{2}$, then $G$ is connected.