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Yes, for $X$ infinite the resulting graph also has chromatic number 2. To see this, just use the fact that a graph is bipartite if and only if it does not contain an odd cycle (this remains true for …

According to this link (see point $15$), Matt DeVos and David Wood can prove that every digraph in which each vertex has an out-neighbour has a good $4$-colouring (I am using their terminology). Up …

The vertex-adding number can be arbitrarily large compared to the chromatic number. To see this consider a long odd cycle, $C_{2k+1}$. Then $\chi(C_{2k+1})=3$, but $a(C_{2k+1})=2k+1$. Note that $ …

No. Suppose that such an $r$ exists. Choose $t \in \mathbb{N}$ such that $\frac{1}{t-1} < r$. Let $G$ be the disjoint union of $t-1$ copies of $K_{t}$. Then $|V(G)|=(t-1)t$ and $\chi(G)=t$, so $\fr …

No, not every graph in your class is planar. Let $G$ be the graph obtained from $K_{3,3}$ by adding a new vertex that is adjacent to all vertices of $K_{3,3}$. Since, $K_{3,3}$ is triangle free, $G$ …

Define $K_n'$ to be the graph obtained from the complete graph on $n$ vertices by subdividing each edge once. Let $G$ be a graph with $\chi(G)=c$ and $\eta(G)=h$. Define $2G$ to be the disjoint union …

It is well known that there is a bijection between the set of trees on $n$ nodes and sequences of length $n-2$ with values in $[n]$. These sequences are called Prüfer sequences. Indeed, the wikipedi …

As far as I know, your conjecture is an open problem, because Reed's conjecture is still open for $\omega=2$. For $\omega=2$, your conjecture is essentially equivalent to Reed's conjecture. That is, …

Here is a proof to a related claim that hopefully will give you some ideas. Claim. Let $X$ be an equivalence class of the entanglement relation on $V(G)$. Then for all distinct $u,v \in X$, there …

Yes, such a graph does exist. Let $G$ be obtained from the complete graph $K_{100}$ by adding two non-adjacent vertices $v$ and $w$ such that $|N_G(v)|=|N_G(w)|=50$ and $N_G(v) \cup N_G(w)=V(K_{100}) …

No, this is false already for $k=3$. Let $A$ and $B$ be disjoint sets of size $n$, and let $H$ be the hypergraph with vertex set $A \cup B$, whose hyperedges are all $3$-subsets $e$ of $A \cup B$ suc …

Let $G$ and $H$ be $k$-critical graphs and $G +_h H$ be the Hajós construction applied to $G$ and $H$ with respect to $vw \in E(G)$ and $xy \in E(H)$. Since $G$ and $H$ are $k$-critical, $G-vw$ and $ …

Here is a reduction to the case that the maximum degree of $G$ is equal to the maximum degree of $H$. In fact, for this reduction, we do not even need the assumption that $H$ are $G$ are in a homomorp …

As requested by Jukka Kohonen, I'll turn my comment into an answer. The answer is in general no. If such an embedding exists, then the red subgraph and blue subgraph are both planar. However, every …

I will use the Euler genus, so that I can also talk about non-orientable surfaces. The Euler genus of a sphere with $g$ handles is $2g$ and the Euler genus of a sphere with $g$ crosscaps is $g$. The …