In this section we show how to apply the ESS coloring to construct colorings of complete split graphs avoiding rainbow 2-connected subgraph *H*. As a result we obtain the lower bounds for anti-Ramsey number *a r*(*K*_{n} +
$\begin{array}{}\overline{{K}_{s}}\end{array}$
, *H*) depending on *n*, *s* and *k* = |*V*(*H*)|. As we always refer to ESS colorings in the proofs, we only indicate a procedure of creating a sequence of components. As we know (see Remark 2.1), the ESS coloring avoids 2-connected subgraph.

We start with the graphs which are large in comparison to a complete part of the split graph.

#### Theorem 3.1

*Let n* ≥ 2, *s* ≥ 1, *k* ≥ 3 *and n* + *s* ≥ *k*. *Let H be an arbitrary 2-connected graph on k* ≥ *n* + 1 *vertices. Then there exists an edge-coloring of*
$\begin{array}{}{K}_{n}+\overline{{K}_{s}}\phantom{\rule{thinmathspace}{0ex}}with\phantom{\rule{thinmathspace}{0ex}}\left(\begin{array}{}\genfrac{}{}{0ex}{}{n}{2}\end{array}\right)+(n-1)(k-1-n)+s\end{array}$
*colors avoiding rainbow H*.

#### Proof

The sequence of components is as follows: firstly we take *s*−(*k* − 1 − *n*)*K*_{1} from the empty part and the last component is the remaining complete split graph.
$\begin{array}{}{K}_{n}+\overline{{K}_{k-1-n}}.\end{array}$
Certainly we use
$\begin{array}{}(\begin{array}{}\genfrac{}{}{0ex}{}{n}{2}\end{array})\end{array}$
+ *n*(*k* − 1 − *n*)+*s* − (*k* − 1 − *n*) =
$\begin{array}{}(\begin{array}{}\genfrac{}{}{0ex}{}{n}{2}\end{array})\end{array}$
+(*n* − 1)(*k* − 1 − *n*)+*s* colors. □

We obtain a straightforward corollary.

#### Corollary 3.2

*Let n* ≥ 2, *s* ≥ 1, *k* ≥ 3 *and n*+*s* ≥ *k*. *Let H be an arbitrary 2-connected graph on k* ≥ *n*+1 *vertices. Then a r*
$\begin{array}{}({K}_{n}+\overline{{K}_{s}},H)\ge (\begin{array}{}\genfrac{}{}{0ex}{}{n}{2}\end{array})+(n-1)(k-1-n)+s.\end{array}$

Now we focus on the cases when the order of the graph *H* is comparable with the order of a complete part of the split graph.

#### Theorem 3.3

*Let n* ≥ 2, *k* ≥ 3,
$\begin{array}{}s\le \lfloor \frac{n}{k-2}\rfloor ,r=(n+s)-\lfloor \frac{n+s}{k-1}\rfloor (k-1)\end{array}$
*and n*+*s* ≥ *k*. *Let H be an arbitrary 2-connected graph on k* ≤ *n vertices. Then there exists an edge-coloring of* $\begin{array}{}{K}_{n}+\overline{{K}_{s}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}with\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\lfloor \frac{n+s}{k-1}\rfloor (\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)+\lceil \frac{n+s}{k-1}\rceil -1\end{array}$
*colors avoiding rainbow H*.

#### Proof

The sequence of components is as follows. Firstly, we take *sK*_{k−2} contained in a complete part *K*_{n}. To each clique *K*_{k−2} we add exactly one vertex from the empty part
$\begin{array}{}\overline{{K}_{s}}\end{array}$
. In such a way we obtain *s* components *K*_{k−1}. Then we divide the remaining vertices of a complete part into cliques *K*_{k−1} making next components being *K*_{k−1}. We may obtain a certain remainder clique *K*_{r} as the last component. Note that this procedure is equivalent to division the whole vertex-set of the graph *K*_{n} +
$\begin{array}{}\overline{{K}_{s}}\end{array}$
into cliques of order *k*−1 in such a way that each clique contains at most one vertex from the empty part. It is possible since *s* ≤
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor .\end{array}$
Altogether we use
$\begin{array}{}\lfloor \frac{n+s}{k-1}\rfloor (\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)+\lceil \frac{n+s}{k-1}\rceil -1\end{array}$
colors. □

#### Corollary 3.4

*Let n* ≥ 2, *k* ≥3,
$\begin{array}{}s\le \lfloor \frac{n}{k-2}\rfloor ,r=(n+s)-\lfloor \frac{n+s}{k-1}\rfloor (k-1)\end{array}$
*and n*+*s* ≥ *k*. *Let H be an arbitrary 2-connected graph on k* ≤ *n vertices. Then a r*
$\begin{array}{}({K}_{n}+\overline{{K}_{s}},H)\ge \lfloor \frac{n+s}{k-1}\rfloor (\begin{array}{l}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)+\lceil \frac{n+s}{k-1}\rceil -1.\end{array}$

#### Theorem 3.5

*Let n* ≥ 2, *k* ≥ 3,
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor =\lceil \frac{n}{k-2}\rceil ,s>\lfloor \frac{n}{k-2}\rfloor \end{array}$
*and n* + *s* ≥ *k*. *Let H be an arbitrary 2-connected graph on k* ≤ *n vertices. Then there exists an edge-coloring of*
$\begin{array}{}{K}_{n}+\overline{{K}_{s}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}with\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{n}{k-2}(\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+s-1\end{array}$
*colors avoiding rainbow H*.

#### Proof

The sequence of components is as follows. Firstly, we take *s*−
$\begin{array}{}\frac{n}{k-2}\phantom{\rule{thinmathspace}{0ex}}\end{array}$
components *K*_{1} from the empty part. Then we take
$\begin{array}{}\frac{n}{k-2}\phantom{\rule{thinmathspace}{0ex}}\end{array}$
*K*_{k−2} contained in a complete part *K*_{n}. To each clique *K*_{k−2} we add exactly one vertex from the empty part
$\begin{array}{}\overline{{K}_{s}}\end{array}$.
In such a way we add
$\begin{array}{}\frac{n}{k-2}\phantom{\rule{thinmathspace}{0ex}}\end{array}$
*K*_{k−1} to the sequence of components. Altogether we use
$\begin{array}{}\frac{n}{k-2}(\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\frac{n}{k-2}-1+s-\frac{n}{k-2}=\frac{n}{k-2}(\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+s-1\end{array}$
colors. □

#### Corollary 3.6

*Let n* ≥ 2, *k* ≥ 3,
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor =\lceil \frac{n}{k-2}\rceil ,s>\lfloor \frac{n}{k-2}\rfloor \end{array}$
*and n*+*s* ≥ *k*. *Let H be an arbitrary 2-connected graph on k* ≤ *n vertices. Then a r*
$\begin{array}{}({K}_{n}+\overline{{K}_{s}},H)\ge \frac{n}{k-2}(\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+s-1.\end{array}$

#### Theorem 3.7

*Let n* ≥ 2, *k* ≥ 3,
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor <\lceil \frac{n}{k-2}\rceil ,r=n-\lfloor \frac{n}{k-2}\rfloor (k-2),\lceil \frac{n}{k-2}\rceil \le s\le \lceil \frac{n}{k-2}\rceil +k-r-1\end{array}$
*and n* + *s* ≥ *k*. *Let H be an arbitrary 2-connected graph on k* ≤ *n vertices. Then there exists an edge-coloring of K*_{n} +
$\begin{array}{}\overline{{K}_{s}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}with\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\lfloor \frac{n}{k-2}\rfloor (\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)+r(s-\lfloor \frac{n}{k-2}\rfloor )+\lceil \frac{n}{k-2}\rceil -1\end{array}$
*colors avoiding rainbow H*.

#### Proof

The sequence of components is as follows. Firstly, we take
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor \end{array}$
*K*_{k−2} contained in a complete part *K*_{n}, to each clique *K*_{k−2} we add exactly one vertex from the empty part
$\begin{array}{}\overline{{K}_{s}}\end{array}$
. In such a way we obtain
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor \end{array}$
components *K*_{k−1}. Then we form a complete split graph from remainining vertices as the last component. Altogether we use
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor (\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)+r(s-\lfloor \frac{n}{k-2}\rfloor )+\lceil \frac{n}{k-2}\rceil -1\end{array}$
colors. □

#### Corollary 3.8

*Let n* ≥ 2, *k* ≥ 3,
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor <\lceil \frac{n}{k-2}\rceil ,r=n-\lfloor \frac{n}{k-2}\rfloor (k-2),\lceil \frac{n}{k-2}\rceil \le s\le \lceil \frac{n}{k-2}\rceil +k-r-1\end{array}$
*and n* + *s* ≥*k*. *Let H be an arbitrary 2-connected graph on k* ≤ *n vertices. Then a r*
$\begin{array}{}({K}_{n}+\overline{{K}_{s}},H)\ge \lfloor \frac{n}{k-2}\rfloor (\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)+r(s-\lfloor \frac{n}{k-2}\rfloor )+\lceil \frac{n}{k-2}\rceil -1.\end{array}$

#### Theorem 3.9

*Let n* ≥ 2, *k* ≥ 3,
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor <\lceil \frac{n}{k-2}\rceil ,r=n-\lfloor \frac{n}{k-2}\rfloor (k-2),s>\lceil \frac{n}{k-2}\rceil +k-r-1\end{array}$
*and n*+*s* ≥ *k*. *Let H be an arbitrary 2-connected graph on k* ≤ *n vertices. Then there exists an edge-coloring of K*_{n} +
$\begin{array}{}\overline{{K}_{s}}\end{array}$
*with*
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor (\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)+(r-1)(k-1-r)+s\end{array}$
*colors avoiding rainbow H*.

#### Proof

We color the edges of *K*_{n} +
$\begin{array}{}\overline{{K}_{s}}\end{array}$
as follows. Firstly, we take
$\begin{array}{}(s-(\frac{n}{k-2}+k-1-r))\end{array}$
*K*_{1} out of the empty part, then
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor \end{array}$
*K*_{k−2} contained in a complete part *K*_{n}, to each clique *K*_{k−2} we add exactly one vertex from the empty part
$\begin{array}{}\overline{{K}_{s}}\end{array}$
. In such a way we obtain next
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor \end{array}$
components *K*_{k−1} in the sequence. Finally, we take a complete split graph
$\begin{array}{}{K}_{r}+\overline{{K}_{k-1-r}}\end{array}$
as the last component. Altogether we use
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor (\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)+r(k-1-r)+s-(\lfloor \frac{n}{k-2}\rfloor +k-1-r)+\lceil \frac{n}{k-2}\rceil -1=\lfloor \frac{n}{k-2}\rfloor (\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)+(r-1)(k-1-r)+s\end{array}$
colors. □

#### Corollary 3.10

*Let n* ≥ 2, *k* ≥ 3,
$\begin{array}{}\lfloor \frac{n}{k-2}\rfloor <\lceil \frac{n}{k-2}\rceil ,r=n-\lfloor \frac{n}{k-2}\rfloor (k-2),s>\lceil \frac{n}{k-2}\rceil +k-r-1\end{array}$
*and n*+*s* ≥ *k*. *Let H be an arbitrary 2-connected graph on k* ≤ *n vertices. Then a r*
$\begin{array}{}({K}_{n}+\overline{{K}_{s}},H)\ge \lfloor \frac{n}{k-2}\rfloor (\begin{array}{}\genfrac{}{}{0ex}{}{k-1}{2}\end{array})+\left(\begin{array}{}\genfrac{}{}{0ex}{}{r}{2}\end{array}\right)\end{array}$
+ (*r*−1)(*k*−1−*r*)+*s*.

## 3.1 Optimality and uniqueness

It is difficult to claim that the colorings presented in the previous section are always optimal (looking at the number of colors used). While constructing these colorings, we pay attention only to the order of the 2-connected graph *H* and the fact that it is bridge-free. So it is quite possible that for a particular graph *H* a better coloring can be constructed looking at its structure. Nevertheless, there are cases that the constructions are best possible in that sense.

Consider the simplest 2-connected graph *H* = *K*_{3}. For this graph we have *k* = 3 so by Theorems 3.3 and 3.5 we obtain the existence of the edge-coloring of *K*_{n} +
$\begin{array}{}\overline{{K}_{s}}\end{array}$
avoiding rainbow triangle with *n* + *s* − 1 colors. It is shown [10] that *a r*(*K*_{n} +
$\begin{array}{}\overline{{K}_{s}}\end{array}$
, *K*_{3}) = *n*+*s*−1. It means that the coloring presented in Theorems 3.3 and 3.5 use maximal possible number of colors. Note that these colorings use maximum matchings and possible single vertices as components and EES coloring. Similarly, as in the case of the complete graph considered as the host graph, changing the position of the componet *K*_{1} (if any), or replacing *K*_{2} by 2*K*_{1} we can obtain other colorings with *n*+*s*−1 colors avoiding rainbow *K*_{3}. In other cases we can also create families of different colorings which avoid rainbow 2-connected subgraph. It is interesting if all of these colorings can be described using ESS colorings or maybe other constructions can appear. Another thing is to establish for which graphs presented colorings are optimal in the sense of number of colors used, i.e. which lower bounds for anti-Ramsey numbers are sharp.

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