On twin edge mean colorings of graphs

Let k ≥ 2 be an integer and G be a connected graph of order at least 3. In this paper, we introduce a new neighbor-distinguishing coloring called twin edge mean coloring. A proper edge coloring of G that uses colors from ℕ k = {0,1,…, k − 1} is called a twin k-edge mean coloring of G if it induces a proper vertex coloring of G such that the color of each vertex υ of G is the average of the colors of the edges incident with υ, and is an integer. The minimum k for which G has a twin k-edge mean coloring is called the twin chromatic mean index of G and is denoted by χtm′(G) . First, we establish lower and upper bounds for χtm′(G) under general or more specific assumptions. Then we determine the twin chromatic mean indices of paths, cycles, and stars.


Introduction
Let G = (V, E) be a simple graph. A proper vertex coloring (resp. proper edge coloring) of G is a function from V (resp. E) to a given set of colors such that adjacent vertices (resp. edges) are colored differently. The minimum number of colors needed in a proper vertex coloring (resp. proper edge coloring) of G is called the chromatic number (resp. chromatic index ) of G and is denoted by χ(G) (resp. χ ′ (G)). An edge coloring c of a graph G is called a neighbor-distinguishing edge coloring if it induces a proper vertex coloring c ′ of G.
In recent decades, various neighbor-distinguishing edge colorings have been introduced and studied in the literature. Some of these studies are the works of Chartrand and Zhang [5], Karonski et al. [8], and Chartrand et al. [6]. In [5], Chartrand and Zhang introduced a neighbor-distinguishing edge coloring of a graph which was called the proper sum k-edge coloring. In particular, for a connected graph G of order at least 3, an edge coloring c : E(G) → [k], where k ∈ N, is called a proper sum k-edge coloring of G if c ′ (x) ̸ = c ′ (y) for every pair x, y of adjacent vertices of G where c ′ (v) is the sum of the colors of edges incident for each v ∈ V (G). The minimum k for which a graph G has a proper sum k-edge coloring is called the sum distiguishing index of G and is denoted by sd(G). In [5], Chartrand where E v is the set of all edges of G incident with v, is an integer. If distinct vertices have distinct chromatic means, then the edge coloring c is called a rainbow mean coloring of G.
For a rainbow mean coloring c of a graph G, the maximum vertex color is the rainbow mean index of c and is denoted by rm(c). The rainbow mean index of the graph G itself, denoted by rm(G) is defined as rm(G) = min{rm(c) | c is a rainbow mean coloring of G}.
In their paper, they investigated the rainbow mean indices of paths, cycles, complete graphs, and stars. On the other hand, Hallas et al. [7] investigated the rainbow mean indices of bipartite graphs.
In 2014, Andrews et al. [1] initially studied a relatively new kind of edge coloring that uses colors from Z k and induces a proper vertex coloring. This edge coloring is called the twin edge coloring and is defined as follows: [1] For a connected graph G of order at least 3, a proper k-edge coloring c : E(G) → Z k for some integer k ≥ 2 is called a twin k-edge coloring of G if the induced vertex coloring c ′ : V (G) → Z k defined by where E v is the set of edges of G incident with v, is proper as well. The minimum k for which G has a twin k-edge coloring is the twin chromatic index of G, denoted by χ ′ t (G). In the past years, several studies on twin edge coloring have been published (see ( [2], [3], [9], [10], for example). In [2], Andrews et al. verified the conjecture that the twin chromatic index of G is at most ∆(G) + 2 for several classes of cubic graphs, all permutation graph of C 5 , prisms, and all trees of maximum degree at most 6. On the other hand, in [3], Andrews et al. also verified the said conjecture for several classes of trees such as brooms, double stars, and regular trees. In addition, Rajarajachozhan and Sampathkumar [9] investigated the twin chromatic indices of squares of paths and cycles, and the cartesian product of paths and cycles while Tolentino et al. [10] investigated the twin chromatic indices of some graphs with maximum degree 3. In this paper, we introduce a new concept which combines some of the characteristics of "twin edge colorings" and "rainbow mean colorings". Throughout the paper, all graphs to be considered are simple, finite, undirected, and connected. Basic notions and definitions will follow the book of Bondy and Murty [4], unless stated otherwise.

Twin Chromatic Mean Index
Definition 3. Let G be a connected graph, k ≥ 2 be an integer, and c : The minimum k for which G has a twin k-edge mean coloring is the twin chromatic mean index of G and is denoted by χ ′ tm (G). Since a twin edge mean coloring of G is a proper edge coloring, χ ′ tm (G) ≥ ∆(G). In this paper, the twin chromatic mean indices of paths, cycles, and stars will be discussed.
The following theorem shows that the twin chromatic mean index of a connected graph of order at least 3 exists.
We will show that c is a twin edge mean coloring of G. It is straightforward to see that for any e i , e j ∈ E(G), we have c(e i ) ̸ = c(e j ); hence, c is proper.
Next, we show that the induced vertex coloring c ′ is proper; that is, c ′ (u) ̸ = c ′ (v) for any two adjacent vertices u and v of G. Let deg(u) = r and deg(v) = s with r ≤ s, and let E u = {e i 1 , e i 2 , ..., e ir } and E v = {e j 1 , e j 2 , ..., e js } where 1 ≤ i 1 < i 2 < · · · < i r ≤ m and 1 ≤ j 1 < j 2 < · · · < j s ≤ m. Accordingly, where, by the choice of q, c ′ (u) and c ′ (v) are both positive integers.
Case 1: Suppose r = s. First, suppose that i r ̸ = j r . We may assume that p = i r > j r .
Next, suppose that i r = j r . Then, In either case, we see that c ′ (u) > c ′ (v), hence c ′ is proper.
Since r < s, this implies that 1 r > 1 s . And so, ICCGANT In either case, we see that c ′ is proper.
The following observations will be useful.
Observation 5. Let G be a connected graph of order at least 3. If ∆(G) is even, then is not an integer. Thus, G has no twin k-edge mean coloring. Proof. Let ∆(G) = k. Suppose on the contrary that χ ′ tm (G) = k; that is, G has a twin k-edge mean coloring, say c : E(G) → N k . Then c ′ (u) = (k − 1)/2 = c ′ (v), a contradiction.

Paths, Cycles, and Stars
To illustrate the concept of twin edge mean coloring, we determine the twin chromatic mean indices of paths, cycles, and stars. We begin with paths.
Theorem 7. If P n is a path of order n ≥ 3, then Observe that, if c : E(G) → N k is a twin k-edge mean coloring of G, then c(e ℓ ) and c(e j ) must have the same parity for each 0 ≤ ℓ < j ≤ n − 2. Moreover, for n ≥ 4, c(e i ) ̸ = c(e i+2 ) for each 0 ≤ i ≤ n − 4. Case 1: Suppose n = 3. Since c(e 0 ) and c(e 1 ) must have the same parity for any twin edge mean coloring of G, we can say that χ ′ tm (G) ̸ = 2; so χ ′ tm (G) ≥ 3. We need to show that χ ′ tm (G) ≤ 3. Now, define an edge coloring c : E(G) → N 3 by c(e 0 ) = 0 and c(e 1 ) = 2. One can check that c is a twin 3-edge mean coloring of G. Hence, χ ′ tm (G) ≤ 3. Case 2: Suppose n ≥ 4. Let c : E(G) → N k be a twin k-edge mean coloring of G. By our observations, we conclude that c(e 0 ), c(e 1 ), and c(e 2 ) are three distinct numbers. Without loss of generality, we consider the parity of c(e 0 ). If c(e 0 ) is even, then max{c(e i ) | 0 ≤ i ≤ n − 2} ≥ 4. On the other hand, if c(e 0 ) is odd, then max{c(e i ) | 0 ≤ i ≤ n − 2} ≥ 5. In either case, max{c(e i ) | 0 ≤ i ≤ n − 2} ≥ 4. This implies that k must be at least 5. Therefore, χ ′ tm (G) ≥ 5. We now show that G has a twin 5-edge mean coloring. Define c : By definition, we can easily see that c is proper. Now, for n ≡ 0 (mod 3), we have If n ≡ 1 (mod 3), we have Lastly, if n ≡ 2 (mod 3), we have In any case, the induced vertex coloring c ′ is proper. Hence, c is a twin 5-edge mean coloring of G.
By definition of c, we can easily observe that c(e i ) and c(e i+1 ) have the same parity, and |{c(e i ), c(e i+1 ), c(e i+2 )}| = 3 for any 0 ≤ i ≤ n − 1. Therefore, c is a twin 7-edge mean coloring of G.   In any case, we see that c(e i ) and c(e i+1 ) have the same parity, and |{c(e i ), c(e i+1 ), c(e i+2 )}| = 3 for any 0 ≤ i ≤ n − 1. Therefore, c is a twin 7-edge mean coloring of G.
Theorem 9. If K 1,s is a star where s ≥ 2, then Proof. Let G = K 1,s be a star, where s ≥ 2. Let V (G) = {v 0 , v 1 , . . . , v s } and E(G) = {e i = v 0 v i | i ∈ {1, 2, . . . , s}}. Since ∆(G) = s, we have χ ′ tm (G) ≥ s. First, we show that χ ′ tm (G) ≥ s + 1. To do that, we need to show that χ ′ tm (G) ̸ = s, that is, G has no twin s-edge mean coloring. By Observation 5, we can assume that s is odd. Suppose on the contrary that χ ′ tm (G) = s; that is, G has a twin s-edge mean coloring c : E(G) → N s . Then c(E(G)) = N s , c ′ (v i ) = c(e i ) for 1 ≤ i ≤ s, and But s−1 2 = c ′ (v i ) for some i ∈ {1, 2, . . . , s}, a contradiction. Thus, χ ′ tm (G) ≥ s + 1. We now consider the cases based on the parity of s. Case 1: Suppose s ≥ 2 is even. We will show that χ ′ tm (G) = s + 1. Since χ ′ tm (G) ≥ s + 1, we only need to show that χ ′ tm (G) ≤ s + 1; that is, that G has a twin (s + 1)-edge mean coloring. If s = 2, then G ∼ = P 3 ; so, by Theorem 7, χ ′ tm (G) = 3 = 2 + 1 when s = 2. We now assume that s ≥ 4. We define an (s + 1)-edge coloring c : E(G) → N s+1 by c(E(G)) = N s+1 \{ s 2 }. It is straightforward to see that c is a proper edge coloring of G. To show that c is a twin (s + 1)-edge mean coloring of G, we need to show that c ′ (v 0 ) is an integer and c ′ (v 0 ) ̸ = c ′ (v i ) for any i ∈ {1, 2, . . . , s}. By definition of c, we have Since s is even, c ′ (v 0 ) = s