2Distance vertexdistinguishing index of subcubic graphs
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Abstract
A 2distance vertexdistinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices at distance 2 have distinct sets of colors. The 2distance vertexdistinguishing index \(\chi ^{\prime }_{\mathrm{d2}}(G)\) of G is the minimum number of colors needed for a 2distance vertexdistinguishing edge coloring of G. Some network problems can be converted to the 2distance vertexdistinguishing edge coloring of graphs. It is proved in this paper that if G is a subcubic graph, then \(\chi ^{\prime }_{\mathrm{d2}}(G)\le 6\). Since the Peterson graph P satisfies \(\chi ^{\prime }_{\mathrm{d2}}(P)=5\), our solution is within one color from optimal.
Keywords
Subcubic graph Edge coloring 2Distance vertexdistinguishing index Starchromatic index1 Introduction
Graph coloring is an important branch of graph theory and combinatorial optimization, because it can be widely applied to various practical problems such as frequency channel assignment, timelabeling and scheduling, planning of experiments, coding and cryptology, etc. The problem in which we are interested is a particular case of the great variety of different ways of labeling a graph. The original motivation of studying this problem came from irregular networks. One of the important tasks in network studies is to be able to identify and distinguish their elements, e.g., vertices, using local substructures and small labels. Among the multiple ways to achieve this, one of the most natural ones is to distinguish vertices by the set of labels in their neighborhoods. Related definitions and results about the problem of distinguishing any two vertices in a graph refer to the literatures (Aigner and Triesch 1990; Balister et al. 2002; Bazgan et al. 1999; Burris 1993; Burris and Schelp 1997). As a special case, a number of authors considered the problem of only distinguishing any pair of adjacent vertices in a graph, see Balister et al. (2007); Hatami (2005); Zhang et al. (2002). The aim of this paper is to deal with another special case in which any two vertices at distance 2 in a graph are required to be distinguish. In fact, this problem also corresponds to a strong application background in networks.
Only simple graphs are considered in this paper. Formally, for a graph G, we use V(G), E(G), and \(\Delta (G)\) (\(\Delta \), for short) to denote its vertex set, edge set, and maximum degree, respectively. A cubic graph is a 3regular graph, and a subcubic graph is a graph of maximum degree at most 3. The distance, denoted by d(u, v), between two vertices u and v is the length of a shortest path connecting them.
A proper edge kcoloring of a graph G is a mapping \(\phi : E(G)\rightarrow \{1,2,\ldots ,k\}\) such that \(\phi (e)\ne \phi (e')\) for any two adjacent edges e and \(e'\). The chromatic index, denoted \(\chi ^{\prime }(G)\), of a graph G is the smallest integer k such that G has a proper edge kcoloring. Let C(v) denote the set of colors assigned to those edges incident to a vertex v, i.e., \(C(v)=\{\phi (uv) uv\in E(G)\}\). The coloring \(\phi \) is called 2distance vertexdistinguishing if \(C(u)\ne C(v)\) for any pair of vertices u and v with \(d (u,v)=2\). The 2distance vertexdistinguishing index \(\chi ^{\prime }_{\mathrm{d2}}(G)\) of a graph G is the smallest integer k such that G has a 2distance vertexdistinguishing edge coloring using k colors.
The 2distance vertexdistinguishing edge coloring of graphs is a special case of the rstrong edge coloring of graphs, which was introduced by Akbari et al. (2006), and independently by Zhang et al. (2006). Let \(r\ge 1\) be an integer. The rstrong chromatic indxe \(\chi ^{\prime }_{\mathrm{s}}(G,r)\) of a graph G is the minimum number of colors required for a proper edge coloring of G such that any two vertices u and v with \(d(u,v)\le r\) have \(C (x)\ne C (y)\).
When \(r=1\), \(\chi ^{\prime }_s(G,1)=\chi ^{\prime }_a(G)\), which is called the neighbordistinguishing chromatic index of G. Note that \(\chi ^{\prime }_{\mathrm{a}}(G)\) can be well defined if and only if G does not contain isolated edges. Zhang et al. (2002) first investigated the neighbordistinguishing edge coloring of graphs and proposed the following conjecture:
Conjecture 1
If G is a connected graph with at least six vertices, then \(\chi ^{\prime }_{\mathrm{a}}(G)\le \Delta +2\).
Balister et al. (2007) comfirmed Conjecture 1 for bipartite graphs and subcubic graphs. Using probabilistic method, Hatami (2005) showed that every graph G with \(\Delta >10^{20}\) has \(\chi ^{\prime }_{\mathrm{a}}(G)\le \Delta +300\). Akbari et al. (2006) proved that every graph G satisfies \(\chi ^{\prime }_{\mathrm{a}}(G)\le 3\Delta \). Wang et al. (2015) improved this result by showing that every graph G has \(\chi ^{\prime }_{\mathrm{a}}(G)\le 2.5\Delta \) if \(\Delta \ge 7\), and \(\chi ^{\prime }_{\mathrm{a}}(G)\le 2\Delta \) if \(\Delta \le 6\). The currently best known upper bound that \(\chi ^{\prime }_{\mathrm{a}}(G)\le 2\Delta +2\) for any graph G was obtained by Vučković (2017).
Wang et al. (2016) first introduced the 2distance vertexdistinguishing edge coloring of graphs and raised the following challenging conjecture:
Conjecture 2
For a graph G, \(\chi ^{\prime }_{\mathrm{d2}}(G)\le \Delta +2\).
It follows from the definition that \(\chi ^{\prime }_{\mathrm{d2}}(G)\ge \chi ^{\prime }(G)\ge \Delta \), and moreover \(\chi ^{\prime }_{\mathrm{d2}}(G)\ge \Delta +1\) if G contains two vertices of maximum degree at distance 2. In Wang et al. (2016), Conjecture 2 was confirmed for some basic graphs such as cycles, paths, trees, complete bipartite graphs, unicycle graphs, and Halin graphs. A graph G is called outerplanar if it can be embedded in the plane such that all vertices lie in the boundary of the unbounded face. Using an algorithmic analysis, Wang et al. (2016) proved that every outerplanar graph G satisfies \(\chi ^{\prime }_{\mathrm{d2}}(G)\le \Delta +8\). In addition, it was shown in Huang et al. (2015) that if G is a bipartite outerplanar graph, then \(\chi ^{\prime }_{\mathrm{d2}}(G)\le \Delta +2\).
A proper edge coloring of a graph G is called staredge coloring if there do not exist bichromatic paths and cycles of length four. That is, at least three colors occur in a path and cycle of length four. The starchromatic index, denoted \(\chi ^{\prime }_{\mathrm{st}}(G)\), of G is the smallest integer k such that G has a staredge kcoloring.
The following relationship is an easy observation, which provides a natural upper bound for \(\chi ^{\prime }_{\mathrm{d2}}(G)\):
Proposition 1
For any graph G, \(\chi ^{\prime }_{\mathrm{d2}}(G)\le \chi ^{\prime }_{\mathrm{st}}(G)\).
More recently, Bezegová et al. (2016) considered the staredge coloring of trees and outerplanar graphs by showing the following results: (i) every tree T has \(\chi ^{\prime }_{\mathrm{st}}(T)\le \lfloor 1.5\Delta \rfloor \), and the upper bound is tight; (ii) every outerplanar graph G has \(\chi ^{\prime }_{\mathrm{st}}(G)\le \lfloor 1.5\Delta \rfloor +12\). Dvořák et al. (2013) proved that every subcubic graph G has \(\chi ^{\prime }_{\mathrm{st}}(G)\le 7\) and conjectured that 7 can be replaced by 6.
Using Proposition 1, we get immediately that \(\chi ^{\prime }_{\mathrm{d2}}(G)\le 7\) for any subcubic graph G. In this paper, we will improve this result by reducing 7 to 6.
2 Main results
Before showing our main results, we introduce a few of concepts and notation. Let G be a graph. A kvertex \((k^\)vertex, \(k^+\)vertex, respectively) of G is a vertex of degree k (at most k, at least k, respectively). For a vertex \(v\in V(G)\), let N(v) denote the set of neighbors of v in G. Let \(\chi (G)\) denote the chromatic number of G, which is the least integer k for which G has a vertex coloring using k colors so that any two adjacent vertices get distinct colors.
In what follows, a 2distance vertexdistinguishing edge kcoloring of G is shortly written as a 2DVDEkcoloring. Suppose that \(\phi \) is a partial 2DVDEcoloring of G using the color set C. We say that two vertices u and v with \(d(u,v)=2\) are conflict under \(\phi \) (or simply conflict) if \(C (u)=C (v)\). An edge uv is said to be legally colored if its color is distinct from the colors of all its neighbors and no pair of conflict vertices at distance 2 are produced.
The following Lemmas 2 and 3 display two famous consequences in graph theory, which will play an important role in the proof of the main result in this paper.
Lemma 2
(Petersen 1891) Every connected cubic graph G without cut edges can be edgepartitioned into a perfect matching and a class of cycles.
Lemma 3
(Brooks 1941) If G is a connected graph and is neither an odd cycle nor a complete graph, then \(\chi (G)\le \Delta \).
Lemma 4
Let G denote the Peterson graph. Then \(\chi ^{\prime }_{\mathrm{d2}}(G)=5\).
Proof
Theorem 5
If G is a subcubic graph, then \(\chi ^{\prime }_{\mathrm{d{2}}}(G)\le 6\).
Proof
Assume that the theorem is not true. Let G be a counterexample with the minimum number of E(G). Then G is connected, and every proper subgraph H of G satisfies \(\chi ^{\prime }_{\mathrm{d{2}}}(H)\le 6\). Suppose that \(\phi \) is a 2DVDE6coloring of a proper subgraph H of G using the color set \(C=\{1,2,\ldots ,6\}\). By showing a series of claims below, we will derive a contradiction.
Claim 1
G does not contain 1vertices.
Proof
Suppose to the contrary that G contains a 1vertex v. Let u be the neighbor of v. Consider the subgraph \(H=Gv\). Then H admits a 2DVDE6coloring \(\phi \) using the color set C by the minimality of G. If \(d(u)=2\), then let w denote the neighbor of u other than v. It suffices to color uv with a color in \(C{\setminus } C(w)\) to extend \(\phi \) to the whole graph G. Otherwise, \(d(u)=3\). Let \(u_1,u_2\) denote the neighbors of u other than v. If at least one of \(u_1\) and \(u_2\) is a \(2^\)vertex, then it suffices to color uv with a color in \(C{\setminus } (C(u_1)\cup C(u_2))\). Otherwise, assume that \(d(u_1)=d(u_2)=3\). Let \(N(u_1)=\{u,w_1,w_2\}\) and \(N(u_2)=\{u,w_3,w_4\}\). Furthermore, we assume that \(\phi (uu_1)=1\) and \(\phi (uu_2)=2\). If uv can not be legally colored, without loss of generality, suppose that \(C(w_i)=\{1,2,i+2\}\) for \(i=1,2,3,4\). If \(C(u_1)=\{1, 3, 4\}\), then we recolor \(uu_1\) with 6 and color uv with 5. Otherwise, by symmetry, we assume \(C(u_1)=\{1, 2, 4\}\). Let \(w'_2\) be the neighbor of \(w_2\) such that \(\phi (w_2w'_2)=2\). If \(4\notin C(w'_2)\), then we recolor \(uu_1\) with 6 and color uv with 5. Otherwise, \(C(w'_2)=\{2, 4, a\}\). Then we recolor \(uu_1\) with a color \(b\in \{5,6\}{\setminus } \{a\}\) and color uv with a color in \(\{5, 6\}{\setminus }\{b\}\). Thus, we always get a 2DVDE6coloring of G, a contradiction. \(\square \)
Claim 2
G does not contain adjacent 2vertices.
Proof
Suppose to the contrary that G contains two adjacent 2vertices u and v with \(N(u)=\{x,v\}\) and \(N(v)=\{u, y\}\). Then \(d(x),d(y)\ge 2\) by Claim 1. If G is a cycle, then G is 2DVDE6colorable by the result in Wang et al. (2016). Now we assume that \(d(x)=3\), for otherwise we consider the neighbor of x until such 3vertex is encountered. By the minimality of G, \(Guv\) admits a 2DVDE6coloring \(\phi \) using the color set C. If \(x=y\), then we color uv with a color in \(C{\setminus } C(x)\). Otherwise, \(x\ne y\). If \(d(y)=2\), then we color uv with a color in \(C{\setminus } (C(x)\cup C(y))\). Otherwise, we assume that \(d(y)=3\). Let \(N(x)=\{u,x_1,x_2\}\) and \(N(y)=\{v,y_1,y_2\}\). If \(C{\setminus } (C(x)\cup C(y))\ge 1\), then we color uv with a color in \(C{\setminus } (C(x)\cup C(y))\). Otherwise, we may assume that \(\phi (ux)=1\), \(\phi (xx_1)=2\), \(\phi (xx_2)=3\), \(\phi (vy)=4\), \(\phi (yy_1)=5\), and \(\phi (yy_2)=6\). If there exists at least one 3vertex in \(\{x_1, x_2, y_1, y_2\}\), say \(d(x_1)=3\), then we color uv with 2. Or else, \(d(x_i)=d(y_i)=2\) for \(i=1,2\). In this case, it suffices to recolor ux with 6 and then color uv with 1. \(\square \)
A 3vertex u is called a \(3_i\)vertex if the number of 2vertices adjacent to u is exactly i.
Claim 3
G does not contain a \(3_{3}\)vertex.
Proof
Suppose to the contrary that G contains a \(3_3\)vertex u. Let \(N(u)=\{u_1, u_2, u_3\}\). Then \(d(u_i)=2\), let \(N(u_{i})=\{u, v_{i}\}\) for \(i=1,2,3\). By Claims 1 and 2, \(v_1,v_2,v_3\) are 3vertices, let \(N(v_1)=\{u_1,v'_1,v''_1\}\), and \(N(v_2)=\{u_2,v'_2,v''_2\}\). By the minimality of G, \(Guu_3\) admits a 2DVDE6coloring \(\phi \) using the color set C. Without loss of generality, assume that \(C(v_3)=\{1,2,3\}\) with \(\phi (u_3v_3)=1\), \(\phi (uu_1)=c_1\), and \(\phi (uu_2)=c_2\).
The proof is split into the following two cases.
Case 1. There is a color \(a\in \{4,5,6\}{\setminus } (C(u_1)\cup C(u_2))\), say \(a=4\).
If \(C(v_1)\ne \{4,c_1,c_2\}\) and \(C(v_2)\ne \{4,c_1,c_2\}\), then we color \(uu_3\) with 4. Otherwise, by symmetry, we may assume that \(C(v_1)=\{4,c_1,c_2\}\). It is easy to see that \(\phi (u_1v_1)=c_2\). We color \(uu_3\) with 4 and then recolor \(uu_1\) with a color \(b\in C{\setminus } \{4,c_1,c_2,\phi (u_2v_2)\}\) such that u does not conflict with \(v_2\).
Case 2. \( 4,5,6\in C(u_1)\cup C(u_2)\).

\(c_2=6\). If \(\phi (u_2v_2)\notin \{4,5\}\), then we recolor \(uu_1\) with a color in \(\{1,2,3\}\) such that \(u_1\) does not conflict with \(v'_1\) and \(v''_1\). Then the proof is reduced to Case 1. So assume that \(\phi (u_2v_2)\in \{4,5\}\). It suffices to recolor \(uu_2\) with a color in \(\{1,2,3\}\) such that \(u_2\) does not conflict with \(v'_2\) and \(v''_2\), and then the proof is reduced to Case 1.

\(c_2\ne 6\). Thus, \(\phi (u_2v_2)=6\). If \(c_2=5\), then we recolor \(uu_1\) with a color in \(\{1,2,3\}\) such that \(u_1\) does not conflict with \(v'_1\) and \(v''_1\) and then the proof is reduced to Case 1. Otherwise, \(c_2\in \{1,2,3\}\). If \(C(v_1)\ne \{4,5,c_2\}\), then we color \(uu_3\) with 5. Otherwise, \(C(v_1)=\{4,5,c_2\}\), say \(\phi (v_1v'_1)=4\) and \(\phi (v_1v''_1)=c_2\). In this case, we color \(uu_3\) with 4, and recolor \(uu_1\) with a color in \(\{1,2,3\}{\setminus } \{c_2\}\).
Claim 4
G does not contain a path \(P=v_0v_1\ldots v_7\) such that \(d(v_1)=d(v_3)=d(v_5)=3\) and \(d(v_2)=d(v_4)=d(v_6)=2\).
Proof
Suppose to the contrary that G contains such a path P, as shown in Fig. 2. For \(i=1,3,5\), we denote by \(u_i\) the neighbor of \(v_i\) other than \(v_{i1}\) and \(v_{i+1}\). By Claim 3, we see that \(d(u_3)=d(u_5)=3\), and there exists at least one 3vertex in \(\{v_0, u_1\}\), say \(d(u_1)=3\). Let \(N(u_3)=\{v_3,x_3,y_3\}\). Let \(H=Gv_2v_3v_3v_4\). By the minimality of G, H has a 2DVDE6coloring \(\phi \) with the color set C such that \(\phi (u_3v_3)=1\), \(\phi (u_3x_3)=2\), and \(\phi (u_3y_3)=3\). Let \(S_1=\{4,5,6\}{\setminus } C(v_1)\) and \(S_5=\{4,5,6\}{\setminus } C(v_5)\). To extend \(\phi \) from H to G, we consider the following cases.
Case 1. \(S_1\ge 1\) and \(S_5\ge 1\).
Let \(a\in S_1\) and \(b\in S_5\). Then the proof splits into two subcases.
Case 1.1. \(a\ne b\), say \(a=4\) and \(b=5\).
Assign 4 to \(v_2v_3\) and 5 to \(v_3v_4\). If \(v_2\) and \(v_4\) are not conflict under the extended coloring, we are done. Otherwise, we derive that \(\phi (v_4v_5)=4\) and \(\phi (v_1v_2)=5\). In this case, if \(6\in S_1\), or \(6\in S_5\), we recolor \(v_2v_3\) or \(v_3v_4\) with 6, and the proof is complete. Otherwise, \(6\notin S_1\cap S_5\). If \(v_3v_4\) can be legally recolored with some color \(c\in \{2,3,6\}\), we are done. Otherwise, \(\phi (v_5v_6)\in \{1, 6\}\), \(C(x_3)= \{1,2,4\}\), and \(C(y_3)=\{1,3,4\}\). We only need recolor \(v_2v_3\) with a color \(c\in \{2,3\}\) such that \(v_2\) does not conflict with \(v_0\).
Case 1.2. \(a=b\).

\(\phi (v_4v_5)=1\). We color \(v_3v_4\) with a color in \(\{2,3,5,6\}\) such that \(v_3\) does not conflict with \(x_3,y_3\), and \(v_4\) does not conflict with \(v_6\).

\(\phi (v_4v_5)=2\). We color \(v_3v_4\) with a color in \(\{5,6\}\) such that \(v_4\) does not conflict with \(v_6\).

\(\phi (v_4v_5)=5\). If \(v_3v_4\) can be legally colored with some color in \(\{2,3,6\}\), we are done. Otherwise, it is immediate to conclude that \(\phi (v_5v_6)=6\), \(\phi (v_6v_7)=5\), \(C(x_3)=\{1,2,4\}\), and \(C(y_3)=\{1,3,4\}\). If \(\phi (v_1v_2)=1\), then we recolor \(v_2v_3\) with 2 and color \(v_3v_4\) with 3. If \(\phi (v_1v_2)\in \{2,3\}\), say \(\phi (v_1v_2)=2\), then we recolor \(v_2v_3\) with 3 and color \(v_3v_4\) with 2. Otherwise, \(\phi (v_1v_2)\in \{5,6\}\), we recolor \(v_2v_3\) with a color \(c\in \{2,3\}\) such that \(v_2\) does not conflict with \(v_0\), and then color \(v_3v_4\) with a color in \(\{2,3\}{\setminus } \{c\}\).
It follows that \(C(v_1)=C(v_5)=\{4,5,6\}\), say \(\phi (v_1v_2)=4\). If neither \(C(x_3)\) nor \(C(y_3)\) is \(\{1,2,3\}\), then we color \(v_2v_3\) with 2 and \(v_3v_4\) with 3. Otherwise, we may assume that \(C(x_3)=\{1,2,3\}\) by symmetry. Assign 3 to \(v_2v_3\). If \(\phi (v_4v_5)\ne \phi (v_6v_7)\), then we color \(v_3v_4\) with a color in \(\{4,5,6\}{\setminus } \{\phi (v_4v_5)\}\) such that \(v_3\) does not conflict with \(y_3\). Otherwise, \(\phi (v_4v_5)=\phi (v_6v_7)\).
By symmetry, we may first assume that \(\phi (v_4v_5)=\phi (v_6v_7)=5\). If \(v_3v_4\) can not be legally colored with any color in \(\{4,6\}\), then \(C(y_3)=\{1,3,4\}\) and \(\phi (v_5v_6)=6\); or \(C(y_3)=\{1,3,6\}\) and \(\phi (v_5v_6)=4\). In this case, we recolor \(v_2v_3\) with 2 and color \(v_3v_4\) with 4 or 6 such that \(v_4\) does not conflict with \(v_6\).
Next assume that \(\phi (v_4v_5)=\phi (v_6v_7)=4\). If \(v_3v_4\) can not be legally colored with any color in \(\{5,6\}\), then \(C(y_3)=\{1,3,5\}\) and \(\phi (v_5v_6)=6\); or \(C(y_3)=\{1,3,6\}\) and \(\phi (v_5v_6)=5\). It suffices to recolor \(v_2v_3\) with 2 and color \(v_3v_4\) with 5 or 6 such that \(v_4\) does not conflict with \(v_6\).
Case 3. \(S_1\ge 1\) and \(S_5=0\).
This means that \(C(v_5)=\{4,5,6\}\). Let \(4\in S_1\). Assign 4 to \(v_2v_3\). If \(v_3v_4\) can not be legally colored with 5 and 6, then \(\phi (v_4v_5)=4\) and \(\phi (v_1v_2)\in \{5, 6\}\), or \(\phi (v_4v_5)\in \{5, 6\}\). Therefore, if \(v_3v_4\) can not be legally colored with 2 and 3, we have \(C(x_3)=\{1, 2, 4\}\) and \(C(y_3)=\{1, 3, 4\}\). If there is \(c\in \{5, 6\}\cap S_1\), then we recolor \(v_2v_3\) with c, and color \(v_3v_4\) with 2 or 3 such that \(v_2\) does not conflict with \(v_4\). Otherwise, \(5,6\in C(v_1)\). If \(\phi (v_1v_2)\in \{2, 3\}\), then we recolor \(v_2v_3\) with \(c\in \{2, 3\}{\setminus }\{\phi (v_1v_2)\}\), and color \(v_3v_4\) with a color in \(\{2,3\}{\setminus } \{c\}\). If \(\phi (v_1v_2)\notin \{2, 3\}\), then we recolor \(v_2v_3\) with \(c\in \{2, 3\}\) such that \(v_2\) does not conflict with \(v_0\), and color \(v_3v_4\) with a color in \(\{2,3\}{\setminus } \{c\}\).
Case 4. \(S_1=0\) and \(S_5\ge 1\).
Note that \(C(v_1)=\{4,5,6\}\). Let \(4\in S_5\), and assign 4 to \(v_3v_4\).
First, assume that \(\phi (v_1v_2)=4\). If \(v_2v_3\) can not be legally colored with 2, 3, 5 or 6, then we can assume that \(C(x_3)=\{1,2,4\}\), \(C(y_3)=\{1,3,4\}\), \(C(v_0)=\{4,5\}\), and \(C(v_4)=\{4,6\}\). Recolor \(v_3v_4\) with a color \(c\in \{2,3\}\) such that \(v_4\) does not conflict with \(v_6\), and then color \(v_2v_3\) with a color in \(\{2,3\}{\setminus } \{c\}\).
Next, assume that \(\phi (v_1v_2)\in \{5,6\}\), say \(\phi (v_1v_2)=5\). If \(v_2v_3\) can not be legally colored with 2, 3, or 6, then we can assume that \(C(x_3)=\{1,2,4\}\), \(C(y_3)=\{1,3,4\}\), and \(C(v_0)=\{5,6\}\). If there is a color \(c\in \{5, 6\}{\setminus } C(v_5)\), then we recolor \(v_3v_4\) with c, and then color \(v_2v_3\) with a color in \(\{2, 3\}\) such that \(v_2\) does not conflict with \(v_4\). Or else, \(5, 6\in C(v_5)\). If \(\phi (v_4v_5)\in \{2, 3\}\), then we recolor \(v_3v_4\) with a color in \(\{2, 3\}{\setminus } \{\phi (v_4v_5)\}\), and then color \(v_2v_3\) with \(\phi (v_4v_5)\). If \(\phi (v_4v_5)\notin \{2, 3\}\), then we recolor \(v_3v_4\) with a color \(c\in \{2,3\}\) such that \(v_4\) does not conflict with \(v_6\), and then color \(v_2v_3\) with a color in \(\{2,3\}{\setminus } \{c\}\). \(\square \)
Claim 5
G does not contain a \(3_{2}\)vertex.
Proof
Suppose to the contrary that G contains a 3vertex u adjacent to two 2vertices \(u_1,u_2\) and a vertex \(u_3\), as shown in Fig. 3. By Claims 1 and 3, \(d(u_3)=3\). Let \(N(u_3)=\{u,x_3,y_3\}\), \(N(u_1)=\{u,v_1\}\), and \(N(u_2)=\{u,v_2\}\). By Claims 1 and 2, \(d(v_i)=3\) for \(i=1,2\). Let \(H=Guu_1uu_2\). By the minimality of G, H has a 2DVDE6coloring \(\phi \) with the color set C such that \(\phi (uu_3)=1\), \(\phi (u_3x_3)=2\), and \(\phi (u_3y_3)=3\). To extend \(\phi \) to G, we have to consider the following two cases.
Case 1. \(v_1=v_2\).
Let \(N(v_1)=\{u_1, u_2, x_1\}\). By Claims 1 and 3, \(x_1\) is a 3vertex. If \(C(v_1)=\{4,5,6\}\) with \(\phi (u_1v_1)=4\) and \(\phi (u_2v_1)=5\), then we color \(uu_1\) with 6 and \(uu_2\) with 4. Otherwise, assume that \(4\notin C(v_1)\), and assign 4 to \(uu_1\). If there is a color \(c\in \{5, 6\}{\setminus } C(v_1)\), then we color \(uu_2\) with c. Otherwise, \(5, 6\in C(v_1)\). When \(\phi (u_2v_1)\in \{2, 3\}\), we derive that \(\phi (u_1v_1)\in \{5, 6\}\), and so color \(uu_2\) with \(\phi (u_1v_1)\). When \(\phi (u_2v_1)\notin \{2, 3\}\), we color \(uu_2\) with a color in \(\{5,6\}{\setminus } \{\phi (u_2v_1)\}\).
Case 2. \(v_1\ne v_2\).
Note that, for \(i=1,2\), each of the other neighbors of \(v_i\), distinct from \(u_i\), are of degree 3 by Claim 4. Let \(T_1=\{4,5,6\}{\setminus } C(v_1)\) and \(T_2=\{4,5,6\}{\setminus } C(v_2)\). By symmetry, we consider three possibilities as follows.
Case 2.1. \(T_1\ge 1\) and \(T_2\ge 1\).
Let \(a\in T_1\) and \(b\in T_2\). First assume that \(a\ne b\), say \(a=4\) and \(b=5\). Assign 4 to \(uu_1\) and 5 to \(uu_2\). If \(C(u_1)\ne C(u_2)\), we are done. Otherwise, we deduce that \(\phi (u_1v_1)=5\) and \(\phi (u_2v_2)=4\). If \(6\in T_1\), then we recolor \(uu_1\) with 6. If \(6\in T_2\), then we recolor \(uu_2\) with 6. So suppose that \(6\in C(v_1)\cap C(v_2)\). If \(C(x_3)\ne \{1,2,4\}\) or \(C(y_3)\ne \{1,3,4\}\), then we recolor \(uu_2\) with 2 or 3. Otherwise, we recolor \(uu_1\) with 2.
Next assume that \(T_1=T_2=\{4\}\), implying that \(5,6\in C(v_1)\cap C(v_2)\). Assign 4 to \(uu_1\). If \(\phi (u_2v_2)\in \{1, 2, 3\}\), then we color \(uu_2\) with 5 or 6. Otherwise, \(\phi (u_2v_2)\in \{5, 6\}\), say \(\phi (u_2v_2)=5\), we color \(uu_2\) with 6.
Case 2.2. \(T_1=T_2=0\).
It follows that \(C(v_1)=C(v_2)=\{4,5,6\}\). Let \(c\in \{4,5,6\} {\setminus } \{\phi (u_1v_1),\phi (u_2v_2)\}\), say \(c=6\). Thus, we can always color \(\{uu_1,uu_2\}\) with two of \(\{2,3,6\}\) such that u does not conflict with \(x_3\) and \(y_3\).
Case 2.3. \(T_1=0\) and \(T_2\ge 1\).
Then \(C(v_1)=\{4,5,6\}\). Let \(4\in T_2\). Assign 4 to \(uu_2\). If \(\phi (u_1v_1)\ne 4\), by symmetry, we can assume that \(\phi (u_1v_1)=5\). Color \(uu_1\) with a color in \(\{2,3,6\}\) such that u does not conflict with \(x_3\) and \(y_3\). Otherwise, \(\phi (u_1v_1)=4\). Then we color \(uu_1\) with some color in \(\{2,3,5,6\}\) such that u does not conflict with \(x_3,y_3\), and \(u_1\) does not conflict with \(u_2\). \(\square \)
Claim 6
No 2vertex lies in a 3cycle.
Proof
Suppose to the contrary that G contains a 2vertex x which lies on the boundary of a 3cycle xyzx. By Claim 2, both y and z are 3vertices. Let \(N(y)=\{x,z,u\}\) and \(N(z)=\{x,y,v\}\). By Claim 5, \(d(u)=d(v)=3\). Let \(N(u)=\{y,u_1,u_2\}\). By the minimality of G, \(Gxy\) admits a 2DVDE6coloring \(\phi \) with the color set C such that \(\phi (xz)=1\), \(\phi (yz)=2\), and \(\phi (vz)=3\).
If \(\phi (uy)=1\), then we color xy with a color in \(\{4,5,6\}\) such that y does not conflict with \(u_1\) and \(u_2\).
If \(\phi (uy)=3\) and xy can not be legally colored with 4, 5, or 6, then we may assume that \(\phi (uy)\in \{4, 5, 6\}\), say \(\phi (uy)=4\). If xy can not be legally colored with 3, 5 or 6, then it is easy to derive, without loss of generality, that \(C(v)=\{2,3,4\}\), \(C(u_1)=\{2,4,5\}\), and \(C(u_2)=\{2,4,6\}\). Switch the colors of xz and yz and then color xy with 3. \(\square \)
Claim 7
G does not contain a 2vertex.
Proof
Using Claims 1–6, we suppose to the contrary that G contains a 2vertex \(u_1\) adjacent to two 3vertices u and \(v_1\). Let \(N(v_1)=\{u_1,x_1,w_1\}\) and \(N(u)=\{u_1,u_2,u_3\}\). By Claim 5, \(d(u_2)=d(u_3)=d(x_1)=d(w_1)=3\). Let \(N(u_2)=\{u,v_2,v_3\}\) and \(N(u_3)=\{u,v_4,v_5\}\). Let \(H=Gu_1+uv_1\). By Claim 6, H is a simple subcubic graph with \(E(H)<E(G)\). By the minimality of G, H admits a 2DVDE6coloring \(\phi \) with the color set C such that \(\phi (uv_1)=1\), \(\phi (uu_2)=2\), \(\phi (uu_3)=3\), \(\phi (x_1v_1)=a\), and \(\phi (w_1v_1)=b\).
To extend \(\phi \) to the whole graph G, we first assign 1 to \(u_1v_1\). If \(uu_1\) can not be legally colored with 4, 5, or 6, then we may assume that \(C(v_i)=\{2,3,i+2\}\) for \(i\in \{2, 3, 4\}\). This implies that \(d(v_i)=3\) for \(i=2,3,4\). Let \(N(v_i)=\{u_2, x_i, w_i\}\) for \(i\in \{2, 3\}\), \(N(v_j)=\{u_3, x_j, w_j\}\) for \(i\in \{4, 5\}\).
If \(C(u_2)=\{2, 4, 5\}\), then we recolor \(uu_2\) with 6 and color \(uu_1\) with 4 or 5 such that u does not conflict with \(v_5\). Otherwise, \(C(u_2)\ne \{2, 4, 5\}\), by symmetry, we assume that \(\phi (u_2v_2)=4\), \(\phi (u_2v_3)=\phi (v_2w_2)=3\), and \(\phi (v_3w_3)=5\). If \(C(w_2)\ne \{1,3,4\}\), then we recolor \(uu_2\) with 1 and color \(uu_1\) with 4, 5, or 6 such that u does not conflict with \(v_1\) and \(v_5\). Now suppose that \(C(w_2)=\{1,3,4\}\). If \(C(w_3)\ne \{3,4,5\}\), then we recolor \(uu_2\) with 5 and color \(uu_1\) with 4 or 6 such that u does not conflict with \(v_5\). Now we furthermore suppose that \(C(w_3)=\{3,4,5\}\). If \(C(u_3)\ne \{3,4,6\}\), then we recolor \(uu_2\) with 6 and color \(uu_1\) with 4 or 5 such that u does not conflict with \(v_5\). Otherwise, \(C(u_3)=\{3,4,6\}\). This implies that \(\phi (u_3v_4)=6\) and \(\phi (u_3v_5)=4\). So assume that \(\phi (v_4x_4)=2\) and \(\phi (v_4w_4)=3\).
Since \(2\le d(v_5)\le 3\), we have to deal with two subcases, depending on the size of \(d(v_5)\).
Case 1. \(d(v_5)=2\).
Let \(N(v_5)=\{u_3,y_5\}\). If \(C(y_5)\ne \{4,5,6\}\), then we recolor \(uu_3\) with 5 and color \(uu_1\) with 4. If \(C(y_5)= \{4,5,6\}\), then we recolor \(uu_3\) with 1 and color \(uu_1\) with a color in \(\{4,5,6\}\) such that u does not conflict with \(v_1\).
Case 2. \(d(v_5)=3\).
If neither \(C(x_5)\) nor \(C(w_5)\) is \(\{4,5,6\}\), then we recolor \(uu_3\) with 5 and color \(uu_1\) with 6. Otherwise, assume that \(C(w_5)=\{4,5,6\}\) by symmetry. If \(C(x_5)\ne \{1,4,6\}\), then we recolor \(uu_3\) with 1 and color \(uu_1\) with 5 or 6 such that u does not conflict with \(v_1\). Otherwise, assume that \(C(x_5)= \{1,4,6\}\). If \(C(x_4)\ne \{2, 4, 6\}\), then we recolor \(uu_3\) with 2 and \(uu_2\) with 6, and then color \(uu_1\) with 5. So assume that \(C(x_4)=\{2, 4, 6\}\).
First assume that \(C(v_5)=\{1,4,5\}\). If \(C(w_4)\ne \{2,3,6\}\), then we recolor \(u_3v_5\) with 2 and \(uu_2\) with 6, and color \(uu_1\) with 4. If \(C(w_4)=\{2,3,6\}\), then we recolor \(u_3v_5\) with 2 and \(uu_3\) with 5, and color \(uu_1\) with 4.

Assume that \(C(w_4)=\{2,3,6\}\). If \(C(z_5)\ne \{1,3,6\}\), then we recolor \(u_3v_5\) with 3 and \(uu_3\) with 5, and color \(uu_1\) with 4. Otherwise, \(C(z_5)=\{1,3,6\}\), we can recolor \(u_3v_5\) with 2 and \(uu_3\) with 5, and color \(uu_1\) with 4.

Assume that \(C(z_5)=\{1,2,6\}\). If \(C(w_4)\ne \{1,3,6\}\), then we recolor \(u_3v_5\) with 3 and \(uu_3\) with 1, and color \(uu_1\) with 4 or 5 such that u does not conflict with \(v_1\). Otherwise, \(C(w_4)=\{1,3,6\}\), we recolor \(u_3v_5\) with 3, \(uu_3\) with 2, \(uu_2\) with 6, and color \(uu_1\) with 5.
Claim 8
G does not contain a cut edge.
Proof
Assume to the contrary that G contains a cut edge uv. Then \(Guv\) consists of exactly two components \(H_1\) and \(H_2\) with \(u\in V(H_1)\) and \(v\in V(H_2)\). Let \(G_1=G[V(H_1)\cup \{v\}]\) and \(G_2=G[V(H_2)\cup \{u\}]\). Then \(G_1\) and \(G_2\) are proper subgraphs of G. By the minimality of G, \(G_1\) has a 2DVDE6coloring \(\phi _1\) using the color set C such that \(C(u)=\{1,2,3\}\) and \(\phi _1(uv)=1\), and \(G_2\) has a 2DVDE6coloring \(\phi _2\) using the same color set C such that \(C(v)=\{1,4,5\}\) and \(\phi _2(uv)=1\). (This can be accomplished by exchanging reasonably the colors of \(G_2\) under \(\phi _2\), if necessarily.) It is easy to inspect that combining \(\phi _1\) and \(\phi _2\) yields a 2DVDE6coloring of G. This is a contradiction. \(\square \)
Case 1. A is good.
It follows that \(L(e_i)\ge 1\) for all \(i=0,1,\ldots , m1\). Let us discuss two subcases below.
Case 1.1. m is even.
We first color \(e_i\) with 1 for \(i=0,2,\ldots ,m2\), and \(e_i\) with 2 for \(i=1,3,\ldots ,m1\). If \(m\equiv 0\)(mod 4), then we recolor \(e_i\) with a color in \(L(e_i)\) for \(i=0,4,8,\ldots ,m4\). If \(m\equiv 2\)(mod 4), then we recolor \(e_i\) with a color in \(L(e_i)\) for \(i=0,4,8,\ldots ,m6\) and moreover recolor \(e_{m3}\) with a color in \(L(e_{m3})\).
Case 1.2. m is odd.
Since m is odd, there must exist a pair of pendant edges \(v_ku_k\) and \(v_{k+2}v_{k+2}\) whose colors are different under \(\pi \), say \(k=m3\). So, we first color \(e_0\) with a color \(a\in L(e_0)\), then color \(e_i\) with 1 for \(i=1,3,\ldots ,m2\), and \(e_i\) with 2 for \(i=2,4,\ldots ,m1\). If \(m\equiv 1\)(mod 4), then we recolor \(e_i\) with a color in \(L(e_i)\) for \(i=4,8,12, \ldots , m5\). If \(m\equiv 3\)(mod 4), then we recolor \(e_i\) with a color in \(L(e_i)\) for \(i=4,8,12, \ldots , m3\).
Case 2. A is bad.

If E(A) is even, then we alternately color the edges of A with 1 and 2 in clockwise order.

If E(A) is odd, then we choose an edge \(e^*\in E(Q_1)\), which is called special edge, and then alternately color the edges of \(E(A){\setminus } \{e^*\}\) with 1 and 2 in clockwise order.
Without loss of generality, assume that that \(Q_i=v_1v_2\cdots v_{k+1}\), i.e., \(Q_i=(e_1,e_2,\ldots ,e_k)\). By the maximality of \(Q_i\), both \(e_0\) and \(e_{k+1}\) are bad. If \(k = 1\), then \(Q_i = \{e_1\}\) and we assign some color in \(L(e_1)\) to \(e_1\). Otherwise, \(k \ge 2\), we define Recoloring Procedures I and II, where (I) runs if \(Q_i\) contains no special edge, and (II) applies only in the case when \(Q_i\) contains a special edge, i.e., A is odd and \(i = 1\).
Recoloring Procedure I.
If \(k\equiv 2\)(mod 4), then we recolor \(e_2\) with a color in \(L(e_2)\), and then recolor \(e_i\) with a color in \(L(e_i)\) for \(i=5,9,13,\cdots ,k1\).
If \(k\equiv 3\)(mod 4), then we recolor \(e_2\) with a color in \(L(e_2)\), and then recolor \(e_i\) with a color in \(L(e_i)\) for \(i=6,10,14,\cdots ,k1\).
If \(k\equiv 0\)(mod 4), then we recolor \(e_2\) with a color in \(L(e_2)\), and then recolor \(e_i\) with a color in \(L(e_i)\) for \(i=6,10,14,\cdots ,k2\).
If \(k\equiv 1\)(mod 4), then we recolor \(e_3\) with a color in \(L(e_3)\), and then recolor \(e_i\) with a color in \(L(e_i)\) for \(i=7,11,15,\cdots ,k2\).
Recoloring Procedure II.
If \(k\not \equiv 1\)(mod 4), then we choose \(e_2\) as special edge and color \(e_2\) with a color in \(L(e_2)\). If \(k\equiv 1\)(mod 4), then we choose \(e_3\) as special edge and color \(e_3\) with a color in \(L(e_3)\). Finally, we recolor other edges (if necessarily) according to Recoloring Procedure I.
Now we come to complete the recoloring for the whole cycle A. If A is an even cycle, then we apply (I) for every path \(Q_i\) for \(i=1,2,\ldots ,r\). If A is an odd cycle, then we apply (I) for every path \(Q_i\) for \(i=2,3,\ldots ,r\) and (II) for \(Q_1\).
It is easy to inspect that the coloring defined above is a 2DVDE6coloring of G. This contradicts the choice of G. \(\square \)
3 Concluding remarks
In this paper, we show that every subcubic graph G satisfies \(\chi ^{\prime }_{\mathrm{d2}}(G)\le 6\). The upper bound value 6 seems not to be best possible. Hence we like to put forward the following probelm:
Conjecture 3
For a subcubic graph G, \(\chi ^{\prime }_{\mathrm{d2}}(G)\le 5\).
Lemma 4 asserts that if Conjecture 3 were true, then the upper bound 5 is tight.
Recall that the proof of Theorem 5 consists of basically two parts: (i) reducing the vertices of degree at most 2 and cut edges; and (ii) dealing with 2connected 3regular graphs. For (ii), we first decompose E(G) into a 1factor M and a 2factor H. Then we establish a strong edge coloring for M, which is restricted in the original graph G, and then extend the coloring of M to the uncolored subgraph H to form a 2DVDE6coloring of the whole graph G. We feel that this coloring scheme can be probably applied to other coloring problems of graphs.
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