Amalgamations and Equitable Block-Colorings

Abstract

An H-decomposition of G is a partition P of E(G) into blocks, each element of which induces a copy of H. Amalgamations of graphs have proved to be a valuable tool in the construction of H-decompositions. The method can force decompositions to satisfy fairness notions. Here the use of the method is further applied to (s, p)-equitable block-colorings of H-decompostions: a coloring of the blocks using exactly s colors such that each vertex v is incident with blocks colored with exactly p colors, the blocks containing v being shared out as evenly as possible among the p color classes. Recently interest has turned to the color vector $V(E)=(c_1(E), c_2(E),$ $\dots , c_s(E))$ of such colorings. Amalgamations are used to construct (s, p)-equitable block-colorings of $C_4$-decompositions of $K_n - F$ and $K_2$-decompositions of $K_{n}$, focusing on one unsolved case with each where $c_1$ is as small as possible and $c_2$ is as large as possible.

1 Introduction

An H-decomposition of a graph G is an ordered pair (V, B) where V is the vertex set of G and B is a partition of the edges of G into sets, each of which induces a copy of H. The elements of B are known as the blocks of the decomposition. An H-decomposition (V, B) is said to have an (s, p)-equitable block-coloring $E: B \mapsto C = \{1,2,\ldots ,s\}$ if:

• (i) the blocks in B are colored with exactly s colors,

• (ii) for each vertex $u\in V(G)$ the blocks containing u are colored using exactly p colors, and

• (iii) for each vertex $u \in V(G)$ and for each $\{i,j\}\subset C(E,u), {\mid } b(E,u,i)-b(E,u,j){\mid } \le 1$,

where $C(E,u)=\{i \mid$ some block incident with u is colored i by $E\}$ and b(E, u, i) is the number of blocks in B containing u that are colored i by E. Such colorings have been considered by several authors, including L. Gionfriddo, M. Gionfriddo, Hork, Li, Matson, Milazzo, Ragusa and Rodger (see [5,6,7, 13, 14]), the work focusing on cases where $H \in \{C_3, C_4\}$ and $G \in \{K_n, K_n - F\}$, where F is a 1-factor of $K_n$. The main focus in these papers was to find the smallest and largest possible values of s for each fixed value of p.

More recently, the research has turned to the structure of such colorings in the form of the color vector $V(E)=(c_1(E), c_2(E),$ $\dots , c_s(E))$ of an (s, p)-equitable block-coloring E of G, where $c_i(E)$ is the number of vertices in G incident with a block of color i arranged in non-decreasing order. Of most interest are the extreme values of $c_i(E)$, thus motivating the following definitions.

Definition 1

For any graphs G and H and for $1\le i\le s$, define

• (i) $\phi (H,G;s,p,i)=\{c_i(E) \mid E$ is an (s, p)-equitable block-coloring of an H-decomposition of G}.

• (ii) $\psi ^{'}(H,G;s,p,i)=\min \phi (H,G;s,p,i)$,

• (iii) and $\overline{\psi ^{'}}(H,G;s,p,i)=\max \phi (H,G;s,p,i)$.

Considering the tightest cases where s is as small as possible for a given value of p is a naturally challenging problem. Often this means that the $s = p$ case is being addressed, and so it is natural to construct such colorings by using path interchange techniques that abound in graph theory. But in rarer cases it turns out that s is always greater than p, requiring new methods to make progress to construct the colorings, hence the motivation for proving Theorems 1 and 2 below. (Interchanging colors along paths can introduce new colors at the end blocks, potentially contravening the requirement that exactly p colors appear on blocks at each vertex.) Before stating Theorem 1, some notation needs to be introduced.

Throughout this paper the focus is on the case where $(H, G) = (C_4, K_{v'} - F)$, and the related case where $(H, G) = (K_2, K_{v})$ described below. In this context it has been shown that the only situation where s is always greater than p is when $v' \equiv 4t+2\,\, (\mathrm {mod}\,{8t})$ (for example, see [14, 15]), in which case if s is as small as possible then $(s,p) = (2t+1, 2t)$ for some integer t. So for the rest of the paper we now assume that $(s,p) = (2t+1, 2t)$ and that $v' = 8tx+4t+2$ for some integer x; so clearly $v'>1$ and $t\ge 1$. It is also convenient to define $\psi ^{'}(H,G;2t+1,2t,i)=\psi ^{'}_i(H,G)$ and $\overline{\psi ^{'}}(H,G;2t+1,2t,i)= \overline{\psi ^{'}}_i(H,G)$. Since each vertex u in $K_{v'} - F$ obviously has degree $8tx+4x$, which is divisible by $2p = 4t$, u is contained in exactly $b'(v') = \frac{v'-2}{4t}=2x+1$ blocks in each of the $p = 2t$ color classes appearing at u (each block, being a 4-cycle, contains 2 edges incident with u). We are now ready to state the following theorem.

Theorem 1

[15]. Let $v'\equiv 4t+2\,\, (\mathrm {mod}\,{8t})$. Let $4t \le 2b'(v') + 2$. Then

• (1) $\psi '_1(C_4,K_{v'}-F)= 2b'(v') + 2$ and

• (2) for $3\le i\le 2t+1$, $\overline{\psi '_{i}}(C_4,K_{v'}-F)=v'-2$.

Notice that there is an unsolved case left in Theorem 1, namely finding $\overline{\psi '_{2}}(C_4,K_{v'}-F)$; this is the one value of i where $\overline{\psi '_{i}}(C_4,K_{v'}-F)$ is not always either the obvious lower or upper bound on the size of a color class, so it is particularly challenging to find. In this paper, $\overline{\psi '_{2}}(C_4,K_{v'}-F)$ is found (see Corollary 1) by solving a related edge-coloring problem in Theorem 4 which is proved using the method of amalgamations of graphs (graph homomorphisms). This construction is then modified in Sect. 4 to provide a new proof of Theorem 1. Amalgamations provide a versatile proof technique that has been used in the study of factorizations of graphs and Steiner triple systems, but its use in block-colorings is relatively new.

Pursuing this approach in more detail, it is shown in [15] that Theorem 1 is a direct consequence of the existence of a $(2t+1,2t)$-equitable edge-coloring of $K_v$, where $v = v'/2$ (or, more precisely, a $(2t+1,2t)$-equitable block-coloring of the obvious $K_2$-decomposition of $K_v$), so $v=4tx+2t+1$ for some integer x; clearly if $v>1$, then $t\ge 1$. Each vertex u has degree $4tx+2t$, which is clearly divisible by $p = 2t$, so u is contained in exactly $b(v) = \frac{v-1}{2t}=2x+1 = b'(v')$ blocks (edges) in each of the 2t color classes appearing at u. Note that b(v) is odd. In Sect. 4, a new proof of the following result is presented (and by the discussion above, a new proof of Theorem 1 as well).

Theorem 2

[15]. Let $v \equiv 2t+1\,\, (\mathrm {mod}\,{4t})$ with $v>1$. Let $2t \le b(v) + 1$. Then,

• (1) $\psi '_1(K_2,K_v)= b(v) + 1$ and

• (2) for $3\le i\le 2t+1$, $\overline{\psi '_{i}}(K_2,K_v) = v-1$.

It is worth noting that a more generalized result in [15] complements Theorems 1 and 2, addressing the cases where $4t\ge 2b'(v')+2$ and $2t\ge b(v)+1$, showing that then $\overline{\psi '_{2}}(C_4,K_{v'}-F)=v'-1$ and $\overline{\psi '_{2}}(K_2,K_v)=v-1$ respectively.

The following notation will be useful throughout the paper. Let K[R] denote the complete graph defined on the vertex set R. Color i is said to appear at a vertex u if at least one block incident with u is colored i.

2 Some Preliminary Results

In order to find $\overline{\psi '_{2}}(C_4,K_{v'}-F)$ and $\overline{\psi '_{2}}(K_2,K_{v})$, we begin by finding bounds on the value of $c_2$ in the following Lemmas, utilizing some results proved in [14, 15]. For ease of notation define $\left\lfloor x \right\rfloor _e$ to be the largest even integer less than or equal to x.

Lemma 1

For $v\equiv 2t+1\,\, (\mathrm {mod}\,{4t})$ and $v'=2v$,

$$\displaystyle {\overline{\psi }'_i(K_2,K_v) \le \left\lfloor \frac{2tv-\sum _{j=1}^{i-1}\psi _j'(K_2,K_v)}{2t+2-i} \right\rfloor _e} \text { and }$$

$$ \displaystyle {\overline{\psi }'_i(C_4,K_{v'}-F) \le \left\lfloor \frac{2tv'-\sum _{j=1}^{i-1}\psi _j'(C_4,K_{v'}-F)}{2t+2-i} \right\rfloor _e}.$$

Proof

Note the elements of the color vector are listed in non-decreasing order; and since in Lemma 2.5 of [14] it is shown that for any $(2t+1,2t)$-equitable edge-coloring E of $K_v$ and for any $(2t+1,2t)$-equitable $C_4$-coloring $E'$ of $K_{v'}-F$, both $\sum _{i=1}^{2t+1} c_i(E) = 2tv$ and $\sum _{i=1}^{2t+1} c_i(E') = 2tv'$, the above holds.   $\square$

Lemma 2

Let $v=v'/2=4tx+2t+1$ for some integer x and $b(v)+1=b'(v')+1\ge 2t$. Then,

$$\overline{\psi }'_2(K_2,K_v)\le \left\lfloor v-\frac{x+1}{t} \right\rfloor _e \text { and }\overline{\psi }'_2(C_4,K_{v'}-F)\le \left\lfloor v'-\frac{2x+2}{t} \right\rfloor _e.$$

Proof

Let $b(v)+1\ge 2t$. By Theorem 3.5 of [15], $\psi '_1(K_2,K_v) = b(v) + 1$. Therefore by Lemma 1:

$$\begin{aligned} \overline{\psi }'_2(K_2,K_v)&\le \lfloor \frac{2tv-(b(v)+1)}{2t+2-2}\rfloor _e\\&= \lfloor v-\frac{(b(v)+1)}{2t}\rfloor _e\\&= \lfloor v-\frac{2x+2}{2t}\rfloor _e\\&= \lfloor v-\frac{x+1}{t}\rfloor _e. \end{aligned}$$

By Corollary 3.6 of [15], $\psi '_1(C_4,K_{v'}-F) = 2b'(v') + 2$. Therefore by Lemma 1:

$$\begin{aligned} \overline{\psi }'_2(C_4, K_{v'}-F)&\le \left\lfloor \frac{2tv'-(2b'(v')+2)}{2t+2-2} \right\rfloor _e\\&= \left\lfloor v'-\frac{(2b'(v')+2)}{2t} \right\rfloor _e\\&= \left\lfloor v'-\frac{2x+4}{2t} \right\rfloor _e\\&= \left\lfloor v'-\frac{2x+2}{t} \right\rfloor _e. \end{aligned}$$

   $\square$

3 Settling the Unsolved Cases in Theorems 1 and 2

Apart from completing the open case left in Theorems 1 and 2, in this paper the use of amalgamations in block-decompositions is further demonstrated. Hilton and Rodger [8, 9] used this technique to find embeddings of edge-colorings into hamiltonian decompositions. Buchanan [2] used amalgamations to find hamiltonian decompositions of $K_{n}-E(U)$ for any 2-regular spanning subgraph U, and this was extended to various multipartite graphs by Leach and Rodger [10, 12]. Leach and Rodger [11] went on to find hamilton decompositions of complete multipartite graphs where each hamilton cycle spreads its edges out as evenly as possible among the pairs of parts of the graph. This notion has recently been extended by Erzurumluoğlu and Rodger [3, 4] to factorizations and holey factorizations of complete multipartite graphs and then to $C_4$-decompositions of $K_v-F$ and edge-decompositions of $K_v$ by Matson and Rodger in [15].

Formally, a graph H is said to be an amalgamation of a graph G if there exists a function $\psi$ from V(G) onto V(H) and a bijection $\psi ^{'}: E(G)\rightarrow E(H)$ such that $e=\lbrace u,v \rbrace \in E(G)$ if and only if $\psi ^{'}(e)=\lbrace \psi (u),\psi (v) \rbrace \in E(H)$. The function $\psi$ is called an amalgamation function. We say that G is a detachment of H, where each vertex u of H splits into the vertices of $\psi ^{-1}(\lbrace u \rbrace )$. An $\eta$-detachment of H is a detachment in which each vertex u of H splits into $\eta (u)$ vertices.

To describe the amalgamation result used here more precisely, some notation will be needed. Let $x\approx y$ represent the fact that $\lfloor y \rfloor \le x\le \lceil y \rceil$. Furthermore, let $\ell (u)$ denote the number of loops incident with vertex u, where each loop contributes twice to the degree of u, let G(j) denote the subgraph of G induced by the edges colored j, and let m(u, v) denote the number of edges between the pair of vertices u and v in G.

The following is a special case of Theorem 3.1 in [1] (omitting the condition that ensures color classes are connected and a balanced property on the color classes for multigraphs since in our case G is simple).

Theorem 3

(Bahmanian and Rodger [1, Theorem 3.1]). Let H be a k-edge-colored graph and let $\eta$ be a function from V(H) into $\mathbb {N}$ such that for each $v \in V(H)$, $\eta (v) = 1$ implies $\ell _H (v) = 0$. Then there exists a loopless $\eta$-detachment G of H in which each $v\in V(H)$ is detached into $v_1,\ldots , v_{\eta (v)}$, such that G satisfies the following conditions:

• 1. $d_G(u_i) \approx d_H(u)/\eta (u)$ for each $u\in V(H)$ and $1\le i\le \eta (u);$

• 2. $d_{G(j)}(u_i) \approx d_{H(j)}(u)/\eta (u)$ for each $u\in V(H)$, $1\le i\le \eta (u)$, and $1\le j\le k;$

• 3. $m_G(u_i, u_{i'}) \approx \ell _H(u)/\left( {\begin{array}{c}\eta (u)\\ 2\end{array}}\right)$ for each $u\in V(H)$ with $\eta (u) \ge 2$ and $1\le i<i'\le \eta (u);$ and

• 4. $m_G(u_i, v_{i'}) \approx m_H(u, v)/(\eta (u) \eta (v))$ for every pair of distinct vertices $u,v\in V(H)$, $1\le i\le \eta (u)$, and $1\le i'\le \eta (v)$.

We now complete the open case left in Theorem 2 as stated here as Theorem 4. As explained in the introduction, as a result of Theorem 4, we also complete the open case left in Theorem 1, stated here as Corollary 1, using the method of amalgamations in both.

Theorem 4

Let $v \equiv 2t+1\,\, (\mathrm {mod}\,{4t})$ with $v>1$. Let $2t \le b(v) + 1$. Then

$$\overline{\psi '_{2}}(K_2,K_v) = \left\lfloor v-\frac{x+1}{t} \right\rfloor _e.$$

Proof

Let $v = 4tx + 2t +1$ for some integer x. Form a complete graph $\mathcal {G}_0$ on the set of vertices $V_0=\{u_1, \dots , u_{2x+2}\}$ and color all the edges of $\mathcal {G}_0$ with color $2t+1$. So each vertex in $\mathcal {G}_0$ is incident with $2x+1 = b(v)$ edges colored $2t+1$ as desired. Notice that in the final edge-coloring of $K_v$, each vertex is missing (i.e., is not incident with any edges of) exactly one color. We will arrange for $1 \le i \le 2t$, color $m(i) = i$ to be missing from vertex $u_i$, for $2t+1\le i\le 2x+2$ color $m(i)= \lceil \frac{i-2t}{2}\rceil \, (\mathrm {mod}\,{2t})\in \{1,\dots , 2t\}$ to be missing from $u_i$, and color $m(\alpha _i)=2t+1$ to be missing from the remaining $v - 2x - 2$ vertices (which will be named $\alpha _1,\ldots , \alpha _{\eta (\alpha )}$ below). For $1\le i\le 2t$ let $M(i)=\{u_j\in V_0\mid m(j)=i\}$. Note for $1\le i<j\le 2t$, $\mid |M(i)| -|M(j)| \mid \in \{0,2\}$ and |M(i)| is odd for all i.

Next form a new edge-colored graph $\mathcal {G}_0^+$ from $\mathcal {G}_0$ as follows. Add a single vertex, $\alpha$. The aim now is to complete the proof by using Theorem 3 with $\eta (u_i) = 1$ for $1 \le i \le 2x+2$ and $\eta (\alpha ) = v - 2x - 2$. For $1 \le i \le 2x+2$ join $u_i$ to $\alpha$ with b(v) edges of each color in $\{1,2, \dots , 2t\} \setminus \{m(i)\}$. Thus for $1\le i\le 2x+2$ the number of edges joining $u_i$ to $\alpha$ is $(2t-1)(2x+1) = 4tx + 2t + 1 - (2x+1) - 1 = v-1-(2x+1) = \eta (\alpha )$, and $d_{\mathcal {G}_0^+}(u_i) = v-1$.

Let a(i) be the number of vertices in $G_0^+$ where color i appears and let $\epsilon _i=2$ for $1\le i\le x+1-t\,\, (\mathrm {mod}\,{2t})$ and $\epsilon _i=0$ otherwise. Therefore $a(2t+1)=2x+2$ and for $1\le i\le 2t$,

$$\begin{aligned} a(i)&= 2x+3-|M(i)|\\&=2x+2-2\left\lfloor \frac{2x+2-2t}{4t} \right\rfloor -\epsilon _i. \end{aligned}$$

Note since $x\ge 0$ and $t\ge 1$ for $1\le i\le 2t$,

$$\begin{aligned} \eta (\alpha )-(a(i)-1)&=v-2x-2-\left( 2x+2-2\left\lfloor \frac{2x+2-2t}{4t} \right\rfloor -\epsilon _i-1\right) \\&=v-4x-3-2\left\lfloor \frac{2x+2-2t}{4t} \right\rfloor +\epsilon _i\\&\ge 4tx+2t-2-4x-\left( \frac{2x+2-2t}{2t}\right) \\&=4x(t-1)+2t-1-\frac{x+1}{t}\\&=(4x+1)(t-1)+t-\frac{x+1}{t}\ge 0. \end{aligned}$$

Thus for $1\le i\le 2t$ add $(b(v)\eta (\alpha )-b(v)(a(i)-1))/2$ loops colored i to $\alpha$, thus resulting in $d_{\mathcal {G}_0^+(i)}(\alpha )=b(v)\eta (\alpha )$. By the above calculations we know we will be adding a non-negative number of loops for all colors $1,\dots , 2t$.

Let $l(\alpha )$ be the number of loops incident with $\alpha$ and $E(V(G_0),\alpha )$ be the set of edges from a vertex in $G_0$ to $\alpha$. Therefore,

$$\begin{aligned} l(\alpha )&= \left( d_{G_0^+}(\alpha )-|E(V(G_0),\alpha )|\right) /2\\&=\left( \eta (\alpha )b(v)2t-(2x+2)[b(v)(2t-1)]\right) /2\\&= \left( \eta (\alpha )b(v)2t-(2x+2)\eta (\alpha )\right) /2\\&=\eta (\alpha )\left( b(v)2t-2x-2\right) /2\\&=\eta (\alpha )\left( 4tx+2t+1-2x-3\right) /2\\&=\eta (\alpha )(v-2x-2-1)/2\\&=\eta (\alpha )(\eta (\alpha )-1)/2. \end{aligned}$$

Now apply Theorem 3 to form the detachment $\mathcal {G}$ of $\mathcal {G}_0^+$ in which $\alpha$ is detached into the vertices $\alpha _1,\ldots , \alpha _{\eta (\alpha )}$. For $1 \le i \le 2x+2$, since $u_i$ is joined to $\alpha$ with b(v) edges in $\mathcal {G}_0^+$, by condition (3) $u_i$ is joined to each vertex $\alpha _j$ for $1 \le j \le \eta (\alpha )$ by exactly one edge in $\mathcal {G}$. Also, since $\alpha$ is incident with $\eta (\alpha )(\eta (\alpha ) -1)/2$ loops in $\mathcal {G}_0^+$, by condition (4) $\alpha _i$ is joined to $\alpha _j$ by exactly one edge for $1 \le i < j \le \eta (\alpha )$ in $\mathcal {G}$. It follows that $\mathcal {G}$ is isomorphic to $K_{2x+2 + \eta (\alpha )} = K_v$. By condition (2), for each vertex u in $\mathcal {G}$, each color which appears at u does so on b(v) edges. Therefore the edge-coloring E of $\mathcal {G}$ is $(2t+1, 2t)$-equitable. Furthermore, in $\mathcal {G}$, color $2t+1$ appears at $b(v) + 1\ge 2t$ vertices and for $1\le i\le 2t$, the number of vertices where color i appears is

$$\begin{aligned} a(i)-1+\eta (\alpha )&=(2x+2)-2\left\lfloor \frac{x+1-t}{2t} \right\rfloor -\epsilon _i-1+v-(2x+2)\\&=v-1-2\left\lfloor \frac{x+1-t}{2t} \right\rfloor -\epsilon _i. \end{aligned}$$

Therefore, since a(i) and $\eta (\alpha )$ are both odd integers, if 2t divides $(x+1-t)$, then $\epsilon _1=0$ and

$$\begin{aligned} a(1)-1 +\eta (\alpha )&=v-1-\frac{x+1-t}{t}\\&=v-\frac{x+1}{t}\\&=\left\lfloor v-\frac{x+1}{t} \right\rfloor _e, \end{aligned}$$

and if 2t does not divide $(x+1-t)$ then $\epsilon _1=2$ and

$$\begin{aligned} a(1)-1+\eta (\alpha )&=v-1-\left( 2\left\lfloor \frac{x+1-t}{2t} \right\rfloor +2\right) \\&=v-1-2\left\lceil \frac{x+1-t}{2t} \right\rceil \\&=v-1+2\left\lfloor \frac{-(x+1-t)}{2t} \right\rfloor \\&=2\left\lfloor \frac{v-1}{2}+\frac{1}{2}-\frac{x+1}{2t} \right\rfloor \\&=2\left\lfloor \frac{v}{2}-\frac{x+1}{2t} \right\rfloor \\&=\left\lfloor v-\frac{x+1}{t} \right\rfloor _e. \end{aligned}$$

Therefore by Lemma 2, $\overline{\psi '_{2}}(K_2,K_v) = \lfloor v-\frac{x+1}{t}\rfloor _e$ and the proof is complete (after renaming color $2t+1$ with 1 and renaming colors $1,2, \dots , 2t$ with $2, 3, \dots ,2t+1$ respectively).    $\square$

Corollary 1

Let $v' \equiv 4t+2 \, (\mathrm {mod}\,{8t})$. Let $2t \le b'(v') + 1$. Then

$$\overline{\psi '_{2}}(C_4, K_{v'}-F) = \lfloor v-\frac{2x+2}{t}\rfloor _e.$$

Proof

By Theorem 4 fo $v=v'/2$ there exists a $(2t+1,2t)$-equitable edge-coloring E of $K_v$ such that $c_2(E)= \lfloor v-\frac{x+1}{t}\rfloor _e$. So as explained in [15] there exists a $(2t+1, 2t)$-equitable $C_4$-coloring $E'$ of $K_{v'}-F$ such that $c_2(E')=2c_2(E)= 2\lfloor v-\frac{x+1}{t}\rfloor _e= \lfloor 2\left( v-\frac{x+1}{t}\right) \rfloor _e=\lfloor 2v-\frac{2x+2}{t}\rfloor _e=\lfloor v'-\frac{2x+2}{t}\rfloor _e$. Therefore by Lemma 2, $\overline{\psi '_{2}}(C_4, K_{v'}-F) = \lfloor v-\frac{2x+2}{t}\rfloor _e$.   $\square$

4 A New Proof of Theorems 1 and 2

By modifying the proof of Theorem 4 we obtain a new proof of Theorem 2, and as explained in the introduction, a new proof of Theorem 1 as well, using amalgamations.

Proof

Let $v = 4tx + 2t +1$ for some integer x. Form $\mathcal {G}_0$ in the same way as in Theorem 4. Here color $m(i) = i$ will be missing from vertex $u_i$ for $1 \le i \le 2t-1$, color $m(i) = 2t$ will be missing from vertex $u_i$ for $2t \le i \le 2x+2$, and color $m(\alpha _i)=2t+1$ will be missing from the remaining $v - 2x - 2$ vertices (which will be named $\alpha _1,\ldots , \alpha _{\eta (\alpha )}$ below).

Next form a new edge-colored graph $\mathcal {G}_0^+$ as in Theorem 4 and again the aim now is to complete the proof using Theorem 3 with $\eta (u_i) = 1$ for $1 \le i \le 2x+2$ and $\eta (\alpha ) = v - 2x - 2$. For $1 \le i \le 2x+2$ join $u_i$ to $\alpha$ with b(v) edges of each color $\{1,2, \dots , 2t\} \setminus \{m(i)\}$ as in Theorem 4; again the number of edges joining $u_i$ to $\alpha$ is $\eta (\alpha )$, and $d_{\mathcal {G}_0^+}(u_i) = v-1$. For $1 \le i \le 2t - 1$ add $b(v)(\eta (v) - (2x+1))/2$ loops of color i to $\alpha$; so $\alpha$ has degree $b(v)\eta (v)$ in color class i (where loops contribute 2 to the degree of the incident vertex). Also add $b(v)(\eta (v) - (2t-1))/2$ loops of color 2t to $\alpha$; so $\alpha$ has degree $b(v)\eta (v)$ in color class 2t as well. Notice that the number of loops incident with $\alpha$ is

$$\begin{aligned} l(\alpha )&=(2t-1)b(v)(\eta (\alpha ) - (2x+1)/2) + b(v)(\eta (\alpha ) - (2t-1))/2\\&=(2t(2x+1)\eta (\alpha ) - (2x+1)(2t-1)(2x+2))/2\\&= (2x+1)(2t\eta (\alpha ) - (4xt - 2x - 4t - 2))/2\\&=(2x+1)(2t\eta (\alpha ) - (\eta (\alpha ) + 2t - 1))/2\\&=(2x+1)(\eta (\alpha ) -1)(2t-1))/2\\&=\eta (\alpha )(\eta (\alpha ) -1)/2. \end{aligned}$$

As in the proof of Theorem 4, Theorem 3 allows us to form $\mathcal {G}$ isomorphic to $K_v$ from $\mathcal {G}_0^+$ so that the edge-coloring E of $\mathcal {G}$ is $(2t+1, 2t)$-equitable. Furthermore, in $\mathcal {G}$, color $2t+1$ appears at $b(v) + 1$ vertices, color 2t appears at $v - 2t -1$ vertices, and each other color appears at $v - 1$ vertices. Since in [15] it is shown in this case that $\psi '_i(K_2,K_v)\ge b(v) + 1$ and that $\overline{\psi '_{i}}(K_2,K_v) \le v-1$ for $1\le i\le 2t+1$, the proof is complete (after renaming the colors $1, 2, \dots , 2t+1$ with $2t +1, 2t, \dots ,1$ respectively).    $\square$