© Wuhan University 2023

0 Introduction

All graphs discussed in this paper are simple, non-directed graphs. Many conclusions have been obtained regarding the vertex-distinguished proper edge coloring^[1-3] and vertex-distinguished general edge coloring^[4-7] of graphs. In 2008, Zhang et al^[8] proposed vertex-distinguished total coloring and related conjectures of graphs. In 2014, Chen et al^[9] introduced vertex-distinguished I-total coloring and related conjectures of graphs. Many studies have been made on vertex-distinguished I-(VI-)total colorings of graphs ^[10-12]. In this study, we consider vertex-distinguished I-(VI-)total colorings of $m{C}_{\mathrm{4}}$ by multiple sets.

Let $G$ be a simple graph. Suppose a mapping $f:V\bigcup E\to \{\mathrm{1,2},\cdots ,l\}$ is a general total coloring of $G$ (not necessarily proper). If $\forall u,v\in V$, and $u,v$ are adjacent vertices, we have $f(u)\ne f(v)$, and if $uv,vw\in$$E,uv\ne vw$, we have $f(uv)\ne f(vw)$, then $f$ is called the I-total coloring of $G$. If any two adjacent edges of $G$ receives different colors, then $f$ is called VI-total coloring of $G$. Obviously, I-total coloring is VI-total coloring, and the reverse is uncertain. For an I- total coloring (resp.VI- total coloring) $f$ of $G$, if $l$ colors are used, then $f$ is called $l$-I-total coloring of $G$ (resp.$l$-VI-total coloring). Note that when we refer to the $l$-I-total coloring (resp.$l$-VI-total coloring) of graph, we always assume that the colors used are $\mathrm{1,2},\cdots ,l$.

Let $f$ be a general total coloring of $G$. For any vertex $x$ in $G$, ${\tilde{C}}_f(x)$ denotes the multiple set of colors of vertex $x$ and edges that are incident of vertex $x$. ${\tilde{C}}_f(x)$ is said to be the color set of $x$ under $f$. No confusion arises when using $\tilde{C}(x)$. Obviously, $|{\tilde{C}}_f(x)|=d_G(x)+\mathrm{1}$. If $\tilde{C}(u)\ne \tilde{C}(v)$ for any two distinct vertices $u$ and $v$ of $G$, then $f$ is called vertex-distinguished by multiple sets. Let

${\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{i}}(G)=\mathrm{m}\mathrm{i}\mathrm{n}${$l|G$ has $l$-I-total coloring which is vertex-distinguished by multiple sets}

and

${\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{v}\mathrm{i}}(G)=\mathrm{m}\mathrm{i}\mathrm{n}${$l|G$ has $l$-VI-total coloring which is vertex-distinguished by multiple sets}.

Then, ${\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{i}}(G)$ is called the I-total chromatic number of $G$ which is vertex-distinguished by multiple sets. Similarly, ${\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{v}\mathrm{i}}(G)$ is called the VI-total chromatic number of $G$ which is vertex-distinguished by multiple sets. Let $n_i(G)$ represent the number of vertices of degree $i$. Suppose that

$\stackrel{}{\tilde{\zeta }}\left(G\right)=\mathrm{m}\mathrm{i}\mathrm{n}\{l|i\left(\begin{array}{l}l\\ i\end{array}\right)+\left(\begin{array}{l}l\\ i+\mathrm{1}\end{array}\right)\ge n_i,\delta \le i\le \mathrm{\Delta }\}$

Proposition 1   ${\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{i}}\left(G\right)\ge {\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{v}\mathrm{i}}\left(G\right)\ge \tilde{\zeta }\left(G\right)$.

Proof   Obviously, I-total coloring is VI-total coloring. Thus ${\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{v}\mathrm{i}}\left(G\right)\le {\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{i}}\left(G\right)$.

Set $t={\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{v}\mathrm{i}}\left(G\right).$$G$ has $t$-VI-total coloring which are vertex-distinguished by multiple sets. For $\delta \le i\le$$\mathrm{\Delta }$. Considering the vertices of the degree $i$, we obtain

$i\left(\begin{array}{l}t\\ i\end{array}\right)+\left(\begin{array}{l}t\\ i+\mathrm{1}\end{array}\right)\ge n_i$

Thus, $t\in \{l|i\left(\begin{array}{l}l\\ i\end{array}\right)+\left(\begin{array}{l}l\\ i+\mathrm{1}\end{array}\right)\ge n_i,\delta \le i\le \mathrm{\Delta }\}$. Therefore, $t\ge \tilde{\zeta }(G)$, namely ${\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{v}\mathrm{i}}\left(G\right)\ge \tilde{\zeta }\left(G\right)$. This completes the proof.

1 Preliminaries

We first define a matrix ${\mathit{A}}_{l\times (l-\mathrm{1})}$, for any $l\ge \mathrm{4}$,

${\mathit{A}}_{l\times (l-\mathrm{1})}=\left(\begin{array}{cccccc}\{\mathrm{1,1},l\}& \{\mathrm{2,2},l\}& \{\mathrm{3,3},l\}& \cdots & \{l-\mathrm{2},l-\mathrm{2},l\}& \{l-\mathrm{1},l-\mathrm{1},l\}\\ \{\mathrm{1,2},l\}& \{\mathrm{2,3},l\}& \{\mathrm{3,4},l\}& \cdots & \{l-\mathrm{2},l-\mathrm{1},l\}& \{l-\mathrm{1},l,l\}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ \{\mathrm{1},l-\mathrm{1},l\}& \{\mathrm{2},l,l\}& \mathrm{\varnothing }& \cdots & \mathrm{\varnothing }& \mathrm{\varnothing }\\ \{\mathrm{1},l,l\}& \mathrm{\varnothing }& \mathrm{\varnothing }& \cdots & \mathrm{\varnothing }& \mathrm{\varnothing }\end{array}\right)$

Let $\mathrm{1}\le i_{\mathrm{1}}<i_{\mathrm{2}}<\cdots <i_r\le l,\mathrm{1}\le j_{\mathrm{1}}<j_{\mathrm{2}}<\cdots <j_s\le l-\mathrm{1}$. Submatrix ${\mathit{A}}_{l\times (l-\mathrm{1})}[i_{\mathrm{1}},i_{\mathrm{2}},\cdots ,i_r|j_{\mathrm{1}},j_{\mathrm{2}},\cdots ,j_s]$ is an$r\times s$ matrix. It is comprised by all the elements which are only in $i_{\mathrm{1}}-,i_{\mathrm{2}}-,\cdots ,$ or $i_r-\mathrm{t}\mathrm{h}$ rows but also in $j_{\mathrm{1}}-,j_{\mathrm{2}}-,\cdots ,$ or $j_s-\mathrm{t}\mathrm{h}$ columns of ${\mathit{A}}_{l\times (l-\mathrm{1})}$. The following six schemes are presented for the I-total coloring of $C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets. Note that all lowercase letters represent different colors.

In Fig. 1(a), the color set of each vertex of $C_{\mathrm{4}}$ is $\{a,a,b\},\{b,b,a\},\{a,a,c\},\{c,c,a\}$. This coloring scheme is Co1$(a;b;c)$.

Fig.1 The coloring scheme Co1$(\mathit{a};\mathit{b};\mathit{c})$, Co2$(\mathit{a},\mathit{b};\mathit{c},\mathit{d};\mathit{e},\mathit{f})$, and Co3$(\mathit{a};\mathit{b};\mathit{c},\mathit{d},\mathit{e})$

In Fig. 1(b), the color set of each vertex of $C_{\mathrm{4}}$ is $\{a,b,c\},\{c,d,a\},\{a,b,e\},\{e,f,a\}$. This coloring scheme is Co2$(a,b;c,d;e,f)$.

In Fig. 1(c), the color set of each vertex of $C_{\mathrm{4}}$ is $\{a,b,b\},\{b,a,a\},\{a,c,d\},\{d,e,a\}$. This coloring scheme is Co3$(a;b;c,d,e)$.

In Fig. 2(a), the color set of each vertex of $C_{\mathrm{4}}$ is $\{a,c,b\},\{b,d,a\},\{a,e,b\},\{b,f,a\}$. This coloring scheme is Co4$(a;b;c;d;e;f)$.

Fig.2 The coloring scheme Co4$(\mathit{a};\mathit{b};\mathit{c};\mathit{d};\mathit{e};\mathit{f})$, Co5$(\mathit{a};\mathit{b};\mathit{c},\mathit{d};\mathit{e};\mathit{f})$, and Co6$(\mathit{a},\mathit{b};\mathit{c};\mathit{d},\mathit{e},\mathit{f})$

In Fig. 2(b), the color set of each vertex of $C_{\mathrm{4}}$ is $\{a,f,b\},\{b,c,d\},\{d,a,e\},\{e,b,a\}$. This coloring scheme is Co5$(a;b;c,d;e;f)$.

In Fig. 2(c), the color set of each vertex of $C_{\mathrm{4}}$ is $\{f,a,b\},\{b,c,a\},\{a,b,d\},\{d,e,f\}$. This coloring scheme is Co6$(a,b;c;d,e,f)$.

Lemma 1   When $\mathrm{1}\le j\le l-\mathrm{2}$ ($j$ is an odd number), $\{j,j,l\},\{j+\mathrm{1},j+\mathrm{1},l\},\{j,l,l\},\{j+\mathrm{1},l,l\}$ are the color sets of the vertices under I-total coloring of $C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets in Fig. 1(a).

Lemma 2   When $i\equiv \mathrm{0}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{2}),j\equiv \mathrm{1}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{2})$, and ${\mathit{A}}_{l\times (l-\mathrm{1})}(i,i+\mathrm{1}|j,j+\mathrm{1})$ are not $\mathrm{\varnothing }$, $\{j,j+i-\mathrm{1},l\}$,$\{j,j+i,$$l\},\{j+\mathrm{1},j+i,l\},\{j+\mathrm{1},j+i+\mathrm{1},l\}$ are the color sets of the vertices under I-total coloring of $C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets in Fig. 1(b).

Lemma 3   If $l\equiv \mathrm{1,6}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{9}$, then for ${\mathit{A}}_{l\times (l-\mathrm{1})}$ except $\mathrm{\varnothing }$ can be divided into $\frac{\mathrm{1}}{\mathrm{4}}\left[\left(\begin{array}{c}l+\mathrm{1}\\ \mathrm{2}\end{array}\right)-\mathrm{1}\right]$ groups, each group has four 3-subsets. These are the color sets of the vertices under I-total coloring of $C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets.

Proof   We use Lemmas 1 and 2, and only consider the remaining entries of ${\mathit{A}}_{l\times (l-\mathrm{1})}$.

Case 1: $l\equiv \mathrm{1}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$.

For each $j\equiv \mathrm{1}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),\mathrm{1}\le j\le l-\mathrm{8}$, considering the remaining entries of the $j,j+\mathrm{2},j+\mathrm{4},j+\mathrm{6}$ columns: $\{j,l-\mathrm{1},l\},\{j+\mathrm{2},l-\mathrm{1},l\},\{j+\mathrm{4},l-\mathrm{1},l\},\{j+\mathrm{6},l-\mathrm{1},l\}$. These four 3-subsets are the color sets of the vertices in Co4$(l;l-\mathrm{1};j;j+\mathrm{2};j+\mathrm{4};j+\mathrm{6})$.

Case 2: $l\equiv \mathrm{6}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$.

Ⅰ. The remaining entries in $\mathrm{1},\mathrm{2},l-\mathrm{3},l-\mathrm{2},l-\mathrm{1}$ columns can be divided into two groups, $\{\{\mathrm{1},l-\mathrm{2},l\},\{l-$$\mathrm{3},l-\mathrm{2},l\},\{l-\mathrm{1},l-\mathrm{1},l\},\{l-\mathrm{1},l,l\}\};\{\{\mathrm{1},l-\mathrm{1},l\},\{\mathrm{2},l-\mathrm{1},l\},\{l-\mathrm{3},$$l-\mathrm{1},l\},\{l-\mathrm{2},l-\mathrm{1},l\}\}$. The corresponding coloring schemes are Co3$(l;l-\mathrm{1};l-\mathrm{3},l-\mathrm{2},\mathrm{1})$ and Co4$(l;l-\mathrm{1};\mathrm{1};$$\mathrm{2};l-\mathrm{3};l-\mathrm{2})$, respectively.

Ⅱ. For each $j\equiv \mathrm{3}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),\mathrm{3}\le j\le l-\mathrm{11}$, considering the remaining entries of the $j,j+\mathrm{1},j+\mathrm{2},j+\mathrm{3},j+\mathrm{4},$$j+\mathrm{5},j+\mathrm{6},j+\mathrm{7}$ columns, which can be divided into three groups: $\{\{j,l-\mathrm{1},l\},\{j+\mathrm{1},l-\mathrm{1},l\},\{j+\mathrm{2},l-\mathrm{1},l\},\{j+\mathrm{3},$

$l-$ $\mathrm{1},l\}\},\{\{j+\mathrm{4},l-\mathrm{1},l\},\{j+\mathrm{5},l-\mathrm{1},l\},\{j+\mathrm{6},l-\mathrm{1},l\},$ $\{j+\mathrm{7},l-\mathrm{1},l\}\},\{\{j,l-\mathrm{2},l\},\{j+\mathrm{2},l-\mathrm{2},l\},$ $\{j+\mathrm{4},l-$ $\mathrm{2},l\},$ $\{j+\mathrm{6},$ $l-\mathrm{2},l\}\}$. The corresponding coloring schemes are Co4$(l;l-\mathrm{1};j;j+\mathrm{1};j+\mathrm{2};j+\mathrm{3})$, Co4$(l;l-\mathrm{1};j+\mathrm{4};j+\mathrm{5};j+\mathrm{6};j+\mathrm{7})$, and Co4$(l;l-\mathrm{2};j;j+\mathrm{2};j+\mathrm{4};j+\mathrm{6})$, respectively.

Lemma 4   If $l\equiv \mathrm{2,5}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{10}$, then all non-empty sets in ${\mathit{A}}_{l\times (l-\mathrm{1})}$ except for $\{l-\mathrm{7},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{2},$$l\}$(when $l\equiv \mathrm{2}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$) or $\{l-\mathrm{4},l-\mathrm{1},l\},\{l-\mathrm{2},l-\mathrm{1},l\}$(when $l\equiv \mathrm{5}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$) can be divided into $\frac{\mathrm{1}}{\mathrm{4}}\left[\left(\begin{array}{c}l+\mathrm{1}\\ \mathrm{2}\end{array}\right)-\mathrm{3}\right]$ groups, and each group has four 3-subsets. These are the color sets of the vertices under I-total coloring of $C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets.

Proof   We use Lemmas 1 and 2, and only consider the remaining entries of ${\mathit{A}}_{l\times (l-\mathrm{1})}$.

Case 1: $l\equiv \mathrm{2}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$.

For the remaining entries in $\mathrm{1,2},l-\mathrm{3},l-\mathrm{2},l-\mathrm{1}$ columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2 Ⅰ. For $j\equiv$$\mathrm{3}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),\mathrm{3}\le j\le l-\mathrm{8}$, considering the remaining entries of the $j,j+\mathrm{1},j+\mathrm{2},j+\mathrm{3},j+\mathrm{4},j+\mathrm{5},j+\mathrm{6},j+\mathrm{7}$ columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2 Ⅱ. For the six remaining entries in $l-\mathrm{7},$$l-\mathrm{6},l-\mathrm{5},l-\mathrm{4}$ columns, there is a group $\{\{l-\mathrm{7},l-\mathrm{1},l\},\{l-\mathrm{6},l-\mathrm{1},l\},\{l-\mathrm{5},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\}\}$, namely Co4$(l;l-\mathrm{1};l-\mathrm{7};l-\mathrm{6};l-\mathrm{5};l-\mathrm{4})$.

This leaves the 3-subsets $\{l-\mathrm{7},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{2},l\}$.

Case 2: $l\equiv \mathrm{5}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$.

For each $j\equiv \mathrm{1}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),\mathrm{1}\le j\le l-\mathrm{5}$, considering the remaining entries in $j,j+\mathrm{2},j+\mathrm{4},j+\mathrm{6}$ columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 1.

This leaves the 3-subsets $\{l-\mathrm{4},l-\mathrm{1},l\},\{l-\mathrm{2},l-\mathrm{1},l\}$.

Lemma 5   If $l\equiv \mathrm{7,0}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{15}$, then all non-empty sets in ${\mathit{A}}_{l\times (l-\mathrm{1})}$ except for $\{l-\mathrm{6},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\},$$\{l-\mathrm{2},l-\mathrm{1},l\}$ (when $l\equiv \mathrm{7}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$) or $\{l-\mathrm{5},l-\mathrm{1},l\},\{l-\mathrm{5},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\}$ (when $l\equiv \mathrm{0}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$) can be divided into $\frac{\mathrm{1}}{\mathrm{4}}\left[\left(\begin{array}{c}l+\mathrm{1}\\ \mathrm{2}\end{array}\right)-\mathrm{4}\right]$ groups, and each group has four 3-subsets. These are the color sets of the vertices under I-total coloring of $C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets.

Proof   We use Lemmas 1 and 2, and only consider the entries of ${\mathit{A}}_{l\times (l-\mathrm{1})}$.

Case 1: $l\equiv \mathrm{7}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$.

For each $j\equiv \mathrm{1}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),\mathrm{1}\le j\le l-\mathrm{7}$, considering the remaining entries in columns $j,j+\mathrm{2},j+\mathrm{4},j+\mathrm{6}$, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 1.

This leaves the 3-subsets $\{l-\mathrm{6},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\},\{l-\mathrm{2},l-\mathrm{1},l\}$.

Case 2: $l\equiv \mathrm{0}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8})$.

For the remaining entries of $\mathrm{1,2},l-\mathrm{3},l-\mathrm{2},l-\mathrm{1}$ columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2 Ⅰ. For $j\equiv \mathrm{3}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),\mathrm{3}\le j\le l-\mathrm{6}$, considering the remaining entries of the $j,j+\mathrm{1},j+$$\mathrm{2},j+\mathrm{3},j+\mathrm{4},j+\mathrm{5},j+\mathrm{6},j+\mathrm{7}$ columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2.

This leaves the 3-subsets $\{l-\mathrm{5},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\}$.

Lemma 6   If $l\equiv \mathrm{3}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{11}$, then all non-empty sets in ${\mathit{A}}_{l\times (l-\mathrm{1})}$ except $\{l-\mathrm{2},l-\mathrm{1},l\}$ can be divided into $\frac{\mathrm{1}}{\mathrm{4}}\left[\left(\begin{array}{c}l+\mathrm{1}\\ \mathrm{2}\end{array}\right)-\mathrm{2}\right]$ groups, and each group has four 3-subsets. These are the color sets of the vertices under I-total coloring of $C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets.

Proof   We use Lemmas 1 and 2, and only consider the remaining entries of ${\mathit{A}}_{l\times (l-\mathrm{1})}$.

For each $j\equiv \mathrm{1}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),\mathrm{1}\le j\le l-\mathrm{3}$, considering the remaining entries of the $j,j+\mathrm{2},j+\mathrm{4},j+\mathrm{6}$ columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 1.

This leaves 3-subset $\{l-\mathrm{2},l-\mathrm{1},l\}$.

Lemma 7   If $l\equiv \mathrm{4}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{12}$, then all non-empty sets in ${\mathit{A}}_{l\times (l-\mathrm{1})}$ except for $\{l-\mathrm{9},l-\mathrm{2},l\},\{l-\mathrm{7},$$l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\}$ can be divided into $\frac{\mathrm{1}}{\mathrm{4}}\left[\left(\begin{array}{c}l+\mathrm{1}\\ \mathrm{2}\end{array}\right)-\mathrm{6}\right]$ groups, and each group has four 3-subsets. These are the color sets of the vertices under I-total colorings of $C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets.

Proof   We use Lemmas 1 and 2, and only consider the remaining entries of ${\mathit{A}}_{l\times (l-\mathrm{1})}$.

For the remaining entries in $\mathrm{1,2},l-\mathrm{3},l-\mathrm{2},l-\mathrm{1}$ columns, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2 Ⅰ. For $j\equiv \mathrm{3}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),\mathrm{3}\le j\le l-\mathrm{10}(l\ge \mathrm{20})$, considering the remaining entries in columns $j,j+\mathrm{1},j+\mathrm{2},j+\mathrm{3},j+\mathrm{4},j+\mathrm{5},j+\mathrm{6},j+\mathrm{7}$, the grouping is obtained and the corresponding coloring scheme is determined using Lemma 3 Case 2. For the remaining in $l-\mathrm{9},l-\mathrm{8},l-\mathrm{7},$$l-\mathrm{6}$ columns, there is a group $\{\{l-\mathrm{9},l-\mathrm{1},l\},\{l-\mathrm{8},l-\mathrm{1},l\},\{l-\mathrm{7},l-\mathrm{1},l\},\{l-\mathrm{6},l-\mathrm{1},l\}\}$ except for $\{l-\mathrm{9},l-\mathrm{2},$$l\},\{l-\mathrm{7},l-\mathrm{2},l\}$, namely Co4$(l;l-\mathrm{1};l-\mathrm{6};l-\mathrm{7};l-\mathrm{8};l-\mathrm{9})$.

This leaves the 3-subsets $\{l-\mathrm{9},l-\mathrm{2},l\},\{l-\mathrm{7},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\}$.

2 Main Results and Their Proofs

Theorem 1   If $\mathrm{2}\left(\begin{array}{c}l-\mathrm{1}\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l-\mathrm{1}\\ \mathrm{3}\end{array}\right)<\mathrm{4}m\le \mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)$$+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right),m\ge \mathrm{1},l\ge \mathrm{3}$, then ${\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{i}}(m{C}_{\mathrm{4}})=l$.

Proof   Obviously, there is $l=\tilde{\zeta }(m{C}_{\mathrm{4}})\le {\tilde{\chi }}_{\mathrm{v}\mathrm{t}}^{\mathrm{i}}(G)$. Therefore, we can directly give the $l$-I-total coloring of $m{C}_{\mathrm{4}}$ which are vertex-distinguished by multiple sets.

① When $m=\mathrm{1},l=\mathrm{3}$. Use $\{\mathrm{1,1},\mathrm{2}\},\{\mathrm{1,2},\mathrm{2}\},\{\mathrm{1,1},\mathrm{3}\},$$\{\mathrm{1,3},\mathrm{3}\}$, that is, Co1$(\mathrm{1};\mathrm{3};\mathrm{2})$ to color the first $C_{\mathrm{4}}$. Thus, the multiple 3-subsets $\{\mathrm{2,2},\mathrm{3}\},\{\mathrm{2,3},\mathrm{3}\},\{\mathrm{1,2},\mathrm{3}\}$ remain.

② When $\mathrm{2}\le m\le \mathrm{4},l=\mathrm{4}$. Based on I-total coloring of the first $C_{\mathrm{4}}$, we start coloring from the second $C_{\mathrm{4}}$. According to Lemma 1, one $C_{\mathrm{4}}$ can be colored with Co1$(\mathrm{4};\mathrm{1};\mathrm{2})$. Subsequently, the multiple 3-subsets $\{\mathrm{3,3},\mathrm{4}\},\{\mathrm{3,4},\mathrm{4}\},\{\mathrm{1,2},\mathrm{4}\},\{\mathrm{1,3},\mathrm{4}\},\{\mathrm{2,3},\mathrm{4}\}$ remain. The third $C_{\mathrm{4}}$ is colored with Co1$(\mathrm{3};\mathrm{4};\mathrm{2})$, under which the color sets of four vertices are $\{\mathrm{3,3},\mathrm{4}\},\{\mathrm{3,4},\mathrm{4}\}$ and ① remaining $\{\mathrm{2,2},\mathrm{3}\},\{\mathrm{2,3},\mathrm{3}\}$. The fourth $C_{\mathrm{4}}$ is colored with ① remaining $\{\mathrm{1,2},\mathrm{3}\}$ and $\{\mathrm{1,2},\mathrm{4}\},\{\mathrm{1,3},\mathrm{4}\},\{\mathrm{2,3},\mathrm{4}\}$, as illustrated in Fig.3. Thus far, all multiple 3-subsets of $\{\mathrm{1,2},\mathrm{3,4}\}$ have been used.

Fig.3 Vertex-distinguished I-total coloring of ${\mathit{C}}_{\mathrm{4}}$

③ When $\mathrm{5}\le m\le \mathrm{7},l=\mathrm{5}$. Based on the previous step, we start coloring from the 5-th $C_{\mathrm{4}}$. According to Lemmas 1 and 2, $\mathrm{3}{C}_{\mathrm{4}}$ can be colored with Co1$(\mathrm{5};\mathrm{1};\mathrm{2})$, Co1$(\mathrm{5};\mathrm{3};\mathrm{4})$, and Co2$(\mathrm{5,3};\mathrm{2,4};\mathrm{1,2})$. Subsequently, the multiple 3-subset $\{\mathrm{1,4},\mathrm{5}\},\{\mathrm{3,4},\mathrm{5}\}$ remain.

④ When $\mathrm{8}\le m\le \mathrm{12},l=\mathrm{6}$. Based on the I-total coloring of $\mathrm{7}{C}_{\mathrm{4}}$ which are vertex-distinguished by multiple sets, we start coloring from the 8-th $C_{\mathrm{4}}$. According to Lemmas 1 and 2, $\mathrm{3}{C}_{\mathrm{4}}$ can be colored with Co1$(\mathrm{6};\mathrm{1};\mathrm{2})$, Co1$(\mathrm{6};\mathrm{3};\mathrm{4})$, and Co2$(\mathrm{6,3};\mathrm{2,4};\mathrm{1,2})$. The remaining entries are $\{\mathrm{5,5},\mathrm{6}\},\{\mathrm{5,6},\mathrm{6}\},\{\mathrm{1,4},\mathrm{6}\},\{\mathrm{1,5},\mathrm{6}\},\{\mathrm{2,5},\mathrm{6}\},\{\mathrm{3,4},\mathrm{6}\},$$\{\mathrm{3,5},\mathrm{6}\},\{\mathrm{4,5},\mathrm{6}\}$, which can be divided into two groups that can color $\mathrm{2}{C}_{\mathrm{4}}$ with Co3$(\mathrm{6};\mathrm{5};\mathrm{1,4},\mathrm{3})$ and Co4$(\mathrm{6};\mathrm{5};\mathrm{1};\mathrm{2};\mathrm{3};\mathrm{4})$. As all multiple 3-subsets $\{\mathrm{1,2},\mathrm{3,4},$$\mathrm{5,6}\}$ containing 6 are used up in the above coloring, the remaining 3-subsets are still $\{\mathrm{1,4},\mathrm{5}\},\{\mathrm{3,4},\mathrm{5}\}$.

⑤ When $\mathrm{13}\le m\le \mathrm{19},l=\mathrm{7}$. Based on the I-total coloring of $\mathrm{12}{C}_{\mathrm{4}}$ which are vertex-distinguished by multiple sets, we color from the 13-th to 19-th $C_{\mathrm{4}}$. According to Lemmas 1 and 2, $\mathrm{6}{C}_{\mathrm{4}}$ can be colored with the Co1$(\mathrm{7};\mathrm{1};\mathrm{2})$, Co1$(\mathrm{7};\mathrm{3};\mathrm{4})$, Co1$(\mathrm{7};\mathrm{5};\mathrm{6})$, Co2$(\mathrm{7,3};\mathrm{2,4};\mathrm{1},$$\mathrm{2})$, Co2$(\mathrm{7,5};\mathrm{2,6};\mathrm{1,4})$, and Co2$(\mathrm{7,5};\mathrm{4,6};\mathrm{3,4})$. Now, the 3-subsets $\{\mathrm{1,6},\mathrm{7}\},\{\mathrm{3,6},\mathrm{7}\},\{\mathrm{5,6},\mathrm{7}\}$ and ③ remaining $\{\mathrm{1,4},\mathrm{5}\}$ are used to color the 19-th $C_{\mathrm{4}}$ with Co6$(\mathrm{7,6};\mathrm{3};\mathrm{1,4},\mathrm{5})$. Due to the above coloring, all multiple 3-subsets containing 7 are used. Thus, the remaining 3-subset $\{\mathrm{3,4},\mathrm{5}\}$ is still left.

⑥ When $\mathrm{20}\le m\le \mathrm{28},l=\mathrm{8}$. We start from the 20-th $C_{\mathrm{4}}$ based on the preceding coloring. According to Lemmas 1 and 2, $\mathrm{6}{C}_{\mathrm{4}}$ can be colored with Co1$(\mathrm{8};\mathrm{1};\mathrm{2})$, Co1$(\mathrm{8};\mathrm{3};\mathrm{4})$, Co1$(\mathrm{8};\mathrm{5};\mathrm{6})$, Co2$(\mathrm{8,3};\mathrm{2,4};\mathrm{1,2})$, Co2$(\mathrm{8,5};\mathrm{2},$$\mathrm{6};\mathrm{1,4})$ and Co2$(\mathrm{8,5};\mathrm{4,6};\mathrm{3,4})$. The remaining 3-subsets are $\{\mathrm{7,7},\mathrm{8}\},\{\mathrm{7,8},\mathrm{8}\},\{\mathrm{1,6},\mathrm{8}\},\{\mathrm{1,7},\mathrm{8}\},\{\mathrm{2,7},\mathrm{8}\},\{\mathrm{3,6},\mathrm{8}\},\{\mathrm{3},$$\mathrm{7},$$\mathrm{8}\},\{\mathrm{4,7},\mathrm{8}\},\{\mathrm{5,6},\mathrm{8}\},\{\mathrm{5,7},\mathrm{8}\},\{\mathrm{6,7},\mathrm{8}\}$. The 3-subsets $\{\mathrm{1,6},\mathrm{8}\},$$\{\mathrm{5,6},\mathrm{8}\},\{\mathrm{7,7},\mathrm{8}\},\{\mathrm{7,8},\mathrm{8}\}$ are used to color the 26-th $C_{\mathrm{4}}$ with Co3$(\mathrm{8};\mathrm{7};\mathrm{1,6},\mathrm{5})$. The 27-th $C_{\mathrm{4}}$ is colored with Co4$(\mathrm{8};\mathrm{7};\mathrm{1};\mathrm{2};\mathrm{5};\mathrm{6})$, under which the color sets of four vertices are $\{\mathrm{1,7},\mathrm{8}\},\{\mathrm{2,7},\mathrm{8}\},\{\mathrm{5,7},\mathrm{8}\},\{\mathrm{6,7},\mathrm{8}\}$. The 3-subsets $\{\mathrm{3,6},\mathrm{8}\},\{\mathrm{3,7},\mathrm{8}\},\{\mathrm{4,7},\mathrm{8}\}$ and ③ remaining $\{\mathrm{3,4},\mathrm{5}\}$ are used to color the 28-th $C_{\mathrm{4}}$ with Co5$(\mathrm{8};\mathrm{3};\mathrm{5,4};\mathrm{7};\mathrm{6})$. Consequently, all multiple 3-subsets of $\{\mathrm{1,2},\mathrm{3},$$\mathrm{4,5},\mathrm{6,7},\mathrm{8}\}$ have been used.

⑦ Let $l\ge \mathrm{9}$, we recursively proceed as following process.

We have obtained ($l-\mathrm{1})$-I-total coloring of $\lfloor \frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l-\mathrm{1}\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l-\mathrm{1}\\ \mathrm{3}\end{array}\right)\right]\rfloor$$C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets. On this basis, we will construct the I-total coloring from the $\lfloor \frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l-\mathrm{1}\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l-\mathrm{1}\\ \mathrm{3}\end{array}\right)\right]\rfloor +\mathrm{1}$-th $C_{\mathrm{4}}$ to $\lfloor \frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)\right]\rfloor$-th $C_{\mathrm{4}}$ which are vertex-distinguished by multiple sets.

When $l\equiv \mathrm{1}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{9}$. Using Lemma 3, we can obtain the $l$-I-total coloring of $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)\right]$$C_{\mathrm{4}}$, which are vertex-distinguished by multiple sets, and we have used all 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$.

When $l\equiv \mathrm{2}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{10}$. Using Lemma 4, we can obtain the $l$-I-total coloring of $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)-\mathrm{2}\right]$$C_{\mathrm{4}}$, which are vertex-distinguished by multiple sets, and we have used all 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$ except for $\{l-$$\mathrm{7},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{2},l\}$.

When $l\equiv \mathrm{3}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{11}$. Using Lemma 6, we can obtain $l$-I-total coloring of $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)-\mathrm{1}\right]C_{\mathrm{4}}$, which are vertex-distinguished by multiple sets. We have used all 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$ except for $\{l-\mathrm{2},l-\mathrm{1},l\}$ and the above mentioned $\{l-\mathrm{8},l-\mathrm{3},l-\mathrm{1}\},\{l-\mathrm{6},l-\mathrm{3},l-\mathrm{1}\}$.

When $l\equiv \mathrm{4}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{12}$. Using Lemma 7, we can obtain the $l$-I-total coloring of $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)-\mathrm{8}\right]C_{\mathrm{4}}$, which are vertex-distinguished by multiple sets. We have used all 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$ except for $\{l-\mathrm{9},l-$$\mathrm{2},l\},\{l-\mathrm{7},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\}$ and the above mentioned $\{l-\mathrm{3},l-\mathrm{2},l-\mathrm{1}\},\{l-\mathrm{9},l-$$\mathrm{4},l-\mathrm{2}\},\{l-\mathrm{7},l-\mathrm{4},l-\mathrm{2}\}$. The $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)-\mathrm{4}\right]$-th $C_{\mathrm{4}}$ is colored with Co2$(l-\mathrm{2},l-\mathrm{4};l-\mathrm{7},l;$$l-\mathrm{9},l)$, under which the color sets of four vertices are $\{l-\mathrm{9},l-\mathrm{2},l\},\{l-\mathrm{7},l-\mathrm{2},l\},\{l-\mathrm{9},l-\mathrm{4},l-\mathrm{2}\},\{l-\mathrm{7},l-\mathrm{4},l-\mathrm{2}\}$. The 3-subsets $\{l-\mathrm{5},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\},\{l-\mathrm{3},l-\mathrm{2},l-\mathrm{1}\}$ are used to color the $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)\right]$-th $C_{\mathrm{4}}$ with Co5$(l;l-\mathrm{1};l-\mathrm{3},l-\mathrm{2};l-\mathrm{5};l-\mathrm{4})$. At this time, we obtained the $l$-I-total coloring of $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)\right]$$C_{\mathrm{4}}$, which are vertex-distinguished by multiple sets. Moreover, all multiple 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$ have been used.

When $l\equiv \mathrm{5}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{13}$. Using Lemma 4, we can obtain the $l$-I-total coloring of $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)-\mathrm{2}\right]$$C_{\mathrm{4}}$, which are vertex-distinguished by multiple sets. We have used all 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$ except for $\{l-\mathrm{4},l-$$\mathrm{1},l\},\{l-\mathrm{2},l-\mathrm{1},l\}$.

When $l\equiv \mathrm{6}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{14}$. Using lemma 3, we can obtain the $l$-I-total coloring of $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)-\mathrm{2}\right]$$C_{\mathrm{4}}$, which are vertex-distinguished by multiple sets. We have used all 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$ except for the above mentioned $\{l-\mathrm{5},l-\mathrm{2},l-\mathrm{1}\},\{l-\mathrm{3},l-\mathrm{2},l-\mathrm{1}\}$.

When $l\equiv \mathrm{7}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{15}$. Using lemma 5, we can obtain the $l$-I-total coloring of $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)-\mathrm{5}\right]$$C_{\mathrm{4}}$, which are vertex-distinguished by multiple sets. We have used all the 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$ except for $\{l-\mathrm{6},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\},\{l-\mathrm{2},l-\mathrm{1},l\}$ and the above mentioned $\{l-\mathrm{6},l-\mathrm{3},l-\mathrm{2}\},\{l-\mathrm{4},$$l-\mathrm{3},l-\mathrm{2}\}$. The 3-subsets $\{l-\mathrm{6},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\},\{l-\mathrm{2},l-\mathrm{1},l\},\{l-\mathrm{6},l-\mathrm{3},l-\mathrm{2}\}$ are used to color the $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)-\mathrm{1}\right]$-th $C_{\mathrm{4}}$ with Co6$(l,l-\mathrm{1};l-\mathrm{4};l-\mathrm{6},l-\mathrm{3},l-\mathrm{2})$. Then the 3-subset $\{l-\mathrm{4},l-\mathrm{3},l-\mathrm{2}\}$ remains.

When $l\equiv \mathrm{0}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{8}),l\ge \mathrm{16}$. Using Lemma 5, we can obtain the $l$-I-total coloring of $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)-\mathrm{4}\right]$$C_{\mathrm{4}}$, which are vertex-distinguished by multiple sets. We have used all the 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$ except for $\{l-$$\mathrm{5},l-\mathrm{2},l\},\{l-\mathrm{5},l-\mathrm{1},l\},\{l-\mathrm{4},l-\mathrm{1},l\}$ and the above mentioned $\{l-\mathrm{5},l-\mathrm{4},l-\mathrm{3}\}$. The $\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{2}\left(\begin{array}{c}l\\ \mathrm{2}\end{array}\right)+\left(\begin{array}{c}l\\ \mathrm{3}\end{array}\right)\right]$-th $C_{\mathrm{4}}$ is colored with the above four 3-subsets, that is, Co5$(l;l-\mathrm{5};l-\mathrm{3},l-\mathrm{4};l-\mathrm{1};l-\mathrm{2})$. Thus far, all multiple 3-subsets of $\{\mathrm{1,2},\cdots ,l\}$ have been used.