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If $X$ is any set, we let $[X]^2:=\big\{\{x,y\}:x\neq y\in X\big\}$. If $G=(V,E)$ is an undirected graph and $v\in V$, we define $N_G(v) = \{w\in V:\{v,w\}\in E\}$.

For $i =1,2$, let $G_i=(V_i,E_i)$ be non-empty, finite, simple, undirected graphs with $V_1\cap V_2 = \emptyset$. Suppose that $E \subseteq [V_1\cup V_2]^2$ has $E\cap [V_i]^2=E_i$ for $i=1,2$, and in addition has the following property:

there is $k\in\mathbb{N}$ such that $|N_G(v_1) \cap V_2| = k$ for all $v_1\in V_1,$

where $G := (V_1\cup V_2, E)$.

Is there a "global constant" $c_k\in\mathbb{N}$ depending on $k$ only such that we have $$\chi(G) \leq \max\{\chi(G_1), \chi(G_2)\} + c_k,$$ no matter what the initial choice of $G_1, G_2$ and $E$ (adhering to the property above) was?

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  • $\begingroup$ Thanks @bof - I will make the notification clearer, and you are right: $2$ is not some "magic" constant: I can do "0 edges connecting $G_1, G_2$", and possibly $1$ edge, but not more. Will generalize to $k$ edges $\endgroup$ Commented Nov 29, 2022 at 12:12
  • $\begingroup$ Thanks @bof, will modify accordingly $\endgroup$ Commented Nov 30, 2022 at 9:03

1 Answer 1

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This is not true even for $k=1$. Assume that $(V_1,E_1)$ is a complete $d$-partite graph with each part having $d$ vertices, and so is $(V_2,E_2)$. Take a bijection $f\colon V_1\to V_2$ and add edges between $v$ and $f(v)$ for all $d^2$ vertices $v\in V_1$. $f$ is chosen so that for each part in $V_1$, its vertices are joined with different parts in $V_2$ (for example, we may join $i$-th vertex of the $j$-th part in $V_1$ with $j$-th vertex of $i$-th part in $V_2$).

Assume that this graph has a proper coloring with $d+c$ colors (where $c=c_1$ is a constant). Note that each color may be used for the vertices of at most one part in $V_1$, and analogously for $V_2$. Therefore at least $d-c$ parts in $G_1$ are monochromatic, and so are at least $d-c$ parts in $G_2$. Thus, if $d>3c$, there exists a color (say, white) such that there is a white part in $V_1$ and white part in $V_2$. But then there is an edge with white endpoints.

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