Adjacent vertex distinguishing colorings by sum of sparse graphs

Discrete Mathematics 339 (2016) 62–71

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Adjacent vertex distinguishing colorings by sum of sparse graphs Xiaowei Yu a , Cunquan Qu a , Guanghui Wang a,∗ , Yiqiao Wang b a

School of Mathematics, Shandong University, 250100, Jinan, Shandong, PR China

b

Academy of Mathematics and System Sciences, Beijing University of Chinese Medicine, 100029, Beijing, PR China

article

info

Article history: Received 22 May 2014 Received in revised form 11 July 2015 Accepted 21 July 2015

Keywords: Proper edge coloring Neighbor sum distinguishing edge coloring Maximum average degree Combinatorial Nullstellensatz

abstract A neighbor sum distinguishing edge-k-coloring, or nsd-k-coloring for short, of a graph G is a proper edge coloring of G with elements from {1, 2, . . . , k} such that no pair of adjacent vertices meets the same sum of colors of G. The definition of this coloring makes sense for graphs containing no isolated edges (we call such graphs normal). Let mad(G) and ∆(G) be the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove that every normal graph with ∆(G) ≥ 5 and mad(G) < 3 admits an nsd-(∆(G) + 2)-coloring. Our approach is based on the Combinatorial Nullstellensatz and the discharging method. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The terminology and notation used but undefined in this paper can be found in [5]. Let G be a finite undirected simple graph. We use V (G), E (G), ∆(G) and δ(G) to denote the vertex set, edge set, maximum degree and minimum degree of the 2|E (H )| graph G, respectively. Let mad(G) = max{ |V (H )| | H ⊂ G} be the maximum average degree of G. Set [n] = {1, 2, . . . , n}, where n is a non-negative integer. A graph G is normal if no connected component is isomorphic to K2 . A proper edge coloring of a graph G = (V (G), E (G)) is an assignment of colors to the edges of G such that no two adjacent edges receive the same color. Let C be a finite set of colors and let c : E (G) → C be a proper edge coloring of G. The color set of a vertex v ∈ V (G) with respect to c is the set S (v) of colors of all edges incident to v , i.e., S (v) = {c (uv) | uv ∈ E (G)}. The coloring c is called a neighbor set distinguishing edge-k-coloring, or an nd-k-coloring of G for short if S (u) ̸= S (v) for each edge uv ∈ E (G). The minimal number of colors necessary to construct such a coloring will be denoted by χa′ (G), and called the neighbor set distinguishing index of G. In 2002, Zhang et al. [22] proposed the following conjecture. Conjecture 1.1 ([22]). If G is a normal connected graph with at least 6 vertices, then χa′ (G) ≤ ∆(G) + 2. Hatami [11] showed that if G is a normal graph and ∆(G) > 1020 , then χa′ (G) ≤ ∆(G) + 300. For more references, see [1,3,8,12]. Recently, in [13], Hocquard et al. proved the following result. Theorem 1.2 ([13]). Let G be a normal graph of maximum degree ∆(G) ≥ 5 and mad(G) < 3 − ∆(2G) , then χa′ (G) ≤ ∆(G) + 1.

∗

Corresponding author. E-mail address: [email protected] (G. Wang).

http://dx.doi.org/10.1016/j.disc.2015.07.011 0012-365X/© 2015 Elsevier B.V. All rights reserved.

X. Yu et al. / Discrete Mathematics 339 (2016) 62–71

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In [20], Wang et al. proved the following result. Theorem 1.3 ([20]). Let G be a normal graph of maximum degree ∆(G) ≥ 3 and mad(G) < 3, then χa′ (G) ≤ ∆(G) + 2. The aim of this paper is to study a similar invariant. Consider  C = [k]. In this case, we are allowed to consider the sum of colors at each vertex v ∈ V (G), i.e., the number w(v) = e∋v c (e). The coloring c is called a neighbor sum distinguishing edge-k-coloring, or an nsd-k-coloring of G for short if w(u) ̸= w(v) for each edge uv ∈ E (G). The least integer k necessary to ′ construct such a coloring will be denoted by χ (G), and called the neighbor sum distinguishing index. Evidently, when searching for the neighbor sum (set) distinguishing index it is sufficient to restrict our attention to connected graphs. Observe also, that a graph G admits a neighbor sum (set) distinguishing edge-k-coloring if and only if G is normal. So, we shall consider only connected graphs with at least three vertices. Apparently, for any normal graph G, ′ ∆(G) ≤ χ ′ (G) ≤ χa′ (G) ≤ χ (G), where χ ′ (G) is the chromatic index of G. A motivation of our work on nsd-k-colorings comes from the study of the following famous conjecture. Conjecture 1.4 ([14], 1-2-3 Conjecture). If G is a graph with no component isomorphic to K2 , then the edges of G may be assigned weights from the set {1, 2, 3} so that, for any adjacent vertices u, v ∈ V (G), the sum of weights of the edges incident to u differs from the sum of weights of the edges incident to v . Recently, Flandrin et al. [9] studied the neighbor sum distinguishing colorings of cycles, trees, complete graphs and complete bipartite graphs. Based on these examples, they proposed the following conjecture. ′ Conjecture 1.5 ([9]). If G is a connected graph on at least 3 vertices and G ̸= C5 , then χ (G) ≤ ∆(G) + 2. ′ Flandrin et al. [9] also proved that for each connected graph G with maximum degree ∆(G) ≥ 2, it holds that χ (G) ≤

⌈ 7∆(G2)−4 ⌉. Wang and Yan [21] improved this bound to ⌈ 10∆(3G)+2 ⌉. In [16], Przybyło proved that ch′ (G) ≤ 2∆(G)+ col(G)− 1, where col(G) is the coloring number of G. Lately, Przybyło and Wong [18] proved that ch′ (G) ≤ ∆(G) + 3col(G) − 4. The ′ latest result thus far is that for any graph with ∆(G) ≥ 2, χ (G) ≤ (1 + o(1))∆(G) [17]. For planar graphs, Dong and ′  Wang [6] proved that χ (G) ≤ max{2∆(G) + 1, 25}. Later this bound was improved to max{∆(G) + 10, 25} by Wang et al. ′ in [19]. In [4], Bonamy and Przybyło proved that any normal planar graph with ∆(G) ≥ 28 satisfies χ (G) ≤ ∆(G)+ 1. Dong et al. also studied neighbor sum distinguishing colorings of sparse graphs in [7]. More precisely, they proved the following result there. Theorem 1.6 ([7]). Let G be a normal graph. If mad(G) < In [10], Gao et al. improved the bound

5 2

to

8 3

5 2

′ and ∆(G) ≥ 5, then χ (G) ≤ ∆(G) + 1.

and proved the following theorem.

Theorem 1.7. Let G be a normal graph. If mad(G) <

8 3

′ and ∆(G) ≥ 5, then χ (G) ≤ ∆(G) + 1.

In this paper, we will prove the following result via the Combinatorial Nullstellensatz and the discharging method. ′ (G) ≤ ∆(G) + 2. Theorem 1.8. Let G be a normal graph. If mad(G) < 3 and ∆(G) ≥ 5, then χ

Apparently, when ∆(G) ≥ 5, Theorem 1.8 implies Theorem 1.3. 2. Preliminaries k

k

Let P (x1 , x2 , . . . , xn ) be a polynomial in n variables, where n ≥ 1. We denote by cP (x11 x22 . . . xknn ) the coefficient of the k k x11 x22

monomial . . . xknn in P (x1 , x2 , . . . , xn ), where ki (1 ≤ i ≤ n) is a non-negative integer. In each configuration of all figures, the degree of each solid vertex is fixed and the degree of each hollow vertex is at least d, where d is the number of solid edges incident with this hollow vertex in the configuration; each solid edge must exist and the existence of every dotted edge cannot be guaranteed. In the following we give several lemmas. Lemma 2.1 ([15]). Let B1 , B2 be sets of integers with |B1 | = m ≥ 2 and |B2 | = n ≥ 2. Let B3 = {x + y | x ∈ B1 , y ∈ B2 , x ̸= y}. Then |B3 | ≥ m + n − 3. Moreover, if B1 ̸= B2 , then |B3 | ≥ m + n − 2.

m

Lemma 2.2 ([15]). Suppose B1 is a set of integers and |B1 | = n. Let B2 = { |B2 | ≥ mn − m2 + 1.

i=1

xi | xi ∈ B1 , xi ̸= xj (i ̸= j)}, where m ≤ n. Then

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Lemma 2.3 (Alon [2], Combinatorial Nullstellensatz). Let Fbe an arbitrary field, and let P = P (x1 , . . . , xn ) be a polynomial n in F[x1 , . . . , xn ]. Suppose the degree deg (P ) of P equals i=1 ki , where each ki is a non-negative integer, and suppose the k

coefficient of x11 ...xknn in P is non-zero. Then if S1 , . . . , Sn are subsets of F with |Si | > ki , there are s1 ∈ S1 , . . . , sn ∈ Sn so that P (s1 , . . . , sn ) ̸= 0. Lemma 2.4. Let

 P ( x1 , x2 , . . . , xn ) =



(xi − xj )

1≤i
n 

2 xk

k =1

be a polynomial in n variables, where n ≥ 2, then cP (xn1 x2n−1 xn3−3 xn4−4 . . . xn−1 ) = 1. Proof. We first show how to compute the coefficient of the monomial xn1 x2n−1 x3n−3 x4n−4 . . . xn−1 . Let

 2  2 n n n    n−2 P1 = ( x1 − xi ) · xj = (x1 − x2 )x1 xj + ··· ; i=2

j=1

j =1

n

P2 =

 (x2 − xi ) = xn2−2 + · · · ; i=3

P3 =



(xi − xj ) = xn3−3 xn4−4 . . . xn−1 + · · · .

3≤i
Observe that P (x1 , x2 , . . . , xn ) = P1 P2 P3 . Denote that P1′ = x1n+1 + 2xn1 x2 − xn1 x2 = xn1+1 + xn1 x2 , P2′ = xn2−2 , P3′ = xn3−3 x4n−4 . . . xn−1 . It is easy to see that the monomial xn1 x2n−1 x3n−3 x4n−4 . . . xn−1 in P only appears in P1′ P2′ P3′ . By easy computations, cP (xn1 x2n−1 xn3−3 xn4−4 . . . xn−1 ) = 1.  3. Proof of Theorem 1.8 Let dG (v) be the degree of a vertex v in G. A vertex v is called a k-vertex (resp. k− -vertex, or k+ -vertex) if dG (v) = k (resp. dG (v) ≤ k, or dG (v) ≥ k). A vertex is called a leaf of the graph G if dG (v) = 1. If u is a 2-vertex of the graph G which is adjacent to a 3− -vertex, we call u a bad 2-vertex. Otherwise, we call u a good 2-vertex. Put k = ∆(G) + 2. Let [k] be the color set, where [k] = {1, 2, . . . , k}. Suppose to the contrary that G is a counterexample to Theorem 1.8, such that |E (G)| + |V (G)| is minimum. In our proofs, we will frequently delete some edges of G to obtain a proper subgraph (not necessarily normal) G′ of G. By the induction hypothesis, there exists an nsd-k-coloring of each normal component of G′ then. If there exist some isolated edges in G′ , then we color them with an arbitrary color in [k]. We call this kind of coloring c of G′ good. For simplicity, let ∆ = ∆ (G) and k = ∆ + 2. We denote by wG (v) the sum of colors taken on the ′ edges incident with v in a coloring of G, i.e. wG (v) = e∋v c (e). To keep things simple, for any proper subgraph G of G, put w(v) = wG′ (v). Let H be the graph obtained by removing all the leaves of the graph G, i.e., H = G − {v ∈ V (G) | dG (v) = 1}. Let N (u) = {v | uv ∈ E (H )}. For a d-vertex u ∈ V (G), if u is adjacent to some 2-vertices ui (1 ≤ i ≤ s ≤ d) and some 3+ -vertices rj , 1 ≤ j ≤ d − s (where rj , 1 ≤ j ≤ d − s might not exist at all), and we denote G′ = G − {uui | 1 ≤ i ≤ s}, then the graph G′ admits a good coloring c. Suppose we wish to extend the coloring c to the whole graph G. Let Si (1 ≤ i ≤ s ≤ d) be the set of available colors for uui . If all the vertices ui (1 ≤ i ≤ s ≤ d) are not pairwise adjacent, where we let NG (ui ) = {u, vi }, then the colors in {c (urj ) | 1 ≤ j ≤ d − s}∪{c (ui vi )}∪{w(vi )− c (ui vi )} (1 ≤ i ≤ s) are forbidden for uui . So |Si | ≥ ∆ + 2 −(d − s + 2) = ∆ − d + s. If some vertex up is adjacent to some vertex uq , where 1 ≤ p ≤ q ≤ s, then the colors in {c (urj ) | 1 ≤ j ≤ d − s} ∪ {c (up uq )} are forbidden for uup and uuq . So |Sp |(|Sq |) ≥ ∆ + 2 −(d − s + 1) = ∆ − d + s + 1. We also can guarantee that wG (up ) ̸= wG (uq ) since c (uup ) ̸= c (uuq ). Thus we just need to consider the case when all the 2-vertices ui (1 ≤ i ≤ s ≤ d) are not pairwise adjacent. 3.1. Case ∆(G) ≥ 9 First, we give some structural properties of the graph H. Claim 3.1. The graph H has the following properties: (i) It holds that δ(H ) ≥ 2. (ii) If dH (u) ≤ 5, then dG (u) = dH (u). (iii) If uv r is a path in H such that dH (v) = 2, 2 ≤ dH (r ) ≤ 5, then dG (u) = dH (u).

X. Yu et al. / Discrete Mathematics 339 (2016) 62–71

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Fig. 1. Illustration of Claim 3.1.

′ Proof. (i) Assume to the contrary that δ(H ) ≤ 1. If δ(H ) = 0, then G is a star K1,∆ . It is thus easy to show that χ (G) = ∆, which is a contradiction to the choice of G. Now we suppose that δ(H ) = 1, then some vertex u is adjacent to at least one leaf z in G, and u is adjacent to exactly one vertex u1 in H, see F1 in Fig. 1. Let G′ = G −{z }, then G′ admits an nsd-k-coloring coloring c. It is clear that we have to exclude at most (∆ − 1) colors taken on all adjacent edges of uz and w(u1 ) − w(u). Hence we have at least ∆ + 2 − (∆ − 1) − 1 = 2 colors for uz to extend the coloring c to G, which is a contradiction. (ii) Denote N (u) = {ui | 1 ≤ i ≤ d}, where d = dH (u) ≤ 5. If the assertion is not true, let z1 , z2 , . . . , zt (1 ≤ t ≤ ∆ − d) be the leaves adjacent to u in the graph G, see F2 in Fig. 1. Let G′ = G − {zj | 1 ≤ j ≤ t }, then G′ admits an nsd-k-coloring c. Case 1: t = 1. Namely, there exists exactly one leaf adjacent to u in G. Notice that the colors in {c (uui ) | 1 ≤ i ≤ d} ∪1≤j≤d {w(uj ) − w(u)} are forbidden for uz1 , so we have at least 11 − 10 = 1 color for uz1 to extend the coloring c to G, which is a contradiction. Case 2: 2 ≤ t ≤ ∆ − d. Observe that the colors in {c (uui ) | 1 ≤ i ≤ d} are forbidden for each uzj (1 ≤ j ≤ t ), so we have at least ∆ + 2 − d available colors for each uzj (1 ≤ j ≤ t ). By Lemma 2.2, if the inequality −t 2 + (∆ + 2 − d)t + 1 ≥ d + 1(∗) holds for all 2 ≤ t ≤ ∆ − d, then we can color all uzj (1 ≤ j ≤ t ) properly so that wG (u) ̸= wG (ui ) for 1 ≤ i ≤ d. It is easy to see that the inequality (∗) holds for t = 2 and t = ∆ − d, thus it holds for 2 ≤ t ≤ ∆ − d by the properties of the quadratic function. Finally, we get an nsd-k-coloring of G, a contradiction. (iii) Let uv r be a path in H such that dH (v) = 2, 2 ≤ dH (r ) ≤ 5, N (r ) = {v, y1 , y2 , . . . , yd−1 }, where 2 ≤ d = dH (r ) ≤ 5. By Claim 3.1(ii), dG (v) = 2, dG (r ) = dH (r ). Suppose that dG (u) > dH (u), and let z1 , z2 , . . . , zt (1 ≤ t ≤ dG (u)− dH (u)) be the leaves adjacent to u in the graph G, see F3 in Fig. 1. Consider G′ = G −{v r }. By the minimality of G, G′ admits a good coloring c. If c (uv) ̸= w(r ), then we color v r with a color α ̸∈ {c (uv)}∪{w(u)− c (uv)}∪{c (ryi ) | 1 ≤ i ≤ d − 1} ∪1≤j≤d−1 {w(yj )− c (ryj )} and we obtain an nsd-k-coloring of the graph G. Otherwise, we can permute the colors assigned to uz1 and uv so that the obtained coloring is still a good coloring of G′ . Now we can extend this coloring to G as previously, a contradiction. This completes the proof of Claim 3.1(iii). 

Claim 3.2. Let u ∈ V (H ), dH (u) = d, uui ∈ E (H ), 1 ≤ i ≤ d. (i) If d = 2, then u is adjacent to at most one 5− -vertex. (ii) If d = 4, then u is adjacent to at most one good 2-vertex. (iii) If 5 ≤ d ≤ ⌈ ∆2 ⌉, then (a) u is not adjacent to any bad 2-vertex. (b) u is adjacent to at most (d − 1) good 2-vertices. (iv)If d ≥ ⌈ ∆2 ⌉ + 1 and u is adjacent to a bad 2-vertex, then u is adjacent to at most (d − 3) 2-vertices. Proof. (i) If the assertion is false, namely, u1 and u2 are two 5− -vertices, then by Claim 3.1(ii), dG (u) = 2, dG (ui ) = dH (ui ) (i = 1, 2). If the vertex u1 and u2 are not adjacent, then for i = 1, 2, denote N (ui ) = {u, ui1 , ui2 , . . . , uid −1 }, where 2 ≤ di ≤ 5, see i

F1 in Fig. 2. Consider G′ = G −{uu1 , uu2 }, then G′ admits a good coloring c. Let Si be the set of available colors for uui (i = 1, 2). It is easy to show that the colors in {c (ui vit ) | 1 ≤ t ≤ di − 1} ∪1≤t ≤di −1 {w(vit ) − w(ui )} ∪ {w(u3−i )} (i = 1, 2) are forbidden for uui , so |Si | ≥ ∆ + 2 − 9 = ∆ − 7 ≥ 2 (i = 1, 2). We can thus color uu1 and uu2 properly and obtain an nsd-k-coloring of G, a contradiction. Now we suppose that the vertex u1 and u2 are adjacent, denote N (ui ) = {u, u3−i , vi1 , . . . , vid −2 }, where 2 ≤ di ≤ 5, i

i = 1, 2. Similarly, consider G′ = G − {uu1 , uu2 }, then G′ admits a good coloring c. Let Si be the set of available colors for uui (i = 1, 2). It is easy to show that the colors in {c (u1 u2 ), c (ui vit ) | 1 ≤ t ≤ di − 2} ∪1≤j≤di −2 {w(vij ) − w(vi )} ∪ {w(u3−i )} (i = 1, 2) are forbidden for uui , so |Si | ≥ ∆ + 2 − 7 = ∆ − 5 ≥ 4 (i = 1, 2). We can thus color uu1 and uu2 properly and so that wG (u1 ) ̸= wG (u2 ). Finally we obtain an nsd-k-coloring of G, a contradiction. Note: According to Claim 3.2(i), all the vertices in {v | dH (v) ≤ 5} are not adjacent to any bad 2-vertex. (ii) Suppose that dH (u1 ) = dH (u2 ) = 2. According to Claim 3.1(ii) and Claim 3.2(i), dG (u1 ) = dG (u2 ) = 2, dG (u) = 4 and u1 , u2 are two good 2-vertices. Let {vi } = N (ui )\{u} (i = 1, 2), see F2 in Fig. 2. Consider G′ = G − {uu1 , uu2 }, then G′ admits a good coloring c. Let Si be the set of available colors for uui (i = 1, 2). Observe that |Si | = ∆ + 2 − |{c (uu3 ), c (uu4 )} ∪ {c (ui vi )} ∪ {c (u3−i v3−i ) − (c (uu3 ) + c (uu4 ))} ∪ {w(vi ) − c (ui vi )}| ≥ ∆ − 3 ≥ 6. According to Lemma 2.1, we can choose αi ∈ Si (i = 1, 2) such that α1 ̸= α2 and wG (u) ̸= wG (uj ) (j = 3, 4). Thus we get an nsd-k-coloring of G, a contradiction.

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X. Yu et al. / Discrete Mathematics 339 (2016) 62–71

Fig. 2. Illustration of Claim 3.2.

(iii) (a) Assume to the contrary that u is adjacent to a bad 2-vertex u1 . Let {v1 } = N (u1 )\{u}. Without loss of generality, suppose that dH (v1 ) = 3. Put {y1 , y2 } = N (v1 )\{u1 }. By Claim 3.1(ii) and (iii), dG (u) = dH (u) = d, dG (u1 ) = 2, dG (v1 ) = dH (v1 ) = 3, see F3 in Fig. 2. Consider G′ = G −{uu1 , u1 v1 }, then G′ admits a good coloring c. Notice that the colors in {c (uui ) | 2 ≤ i ≤ d}∪{w(v1 )} ∪2≤j≤d {w(uj )−w(u)} are forbidden for uu1 . So the number of safe colors for uu1 is at least ∆ + 2 − 2(d − 1)− 1 ≥ ∆ + 2 − 2(⌈ ∆ ⌉− 1)− 1 ≥ 2, thus we may color uu1 with one of these. Now we color u1 v1 with a color α ̸∈ {c (uu1 )}∪ 2 {c (v1 y1 ), c (v1 y2 )} ∪ {w(y1 ) − w(v1 )} ∪ {w(y2 ) − w(v1 )} ∪ {w(u)} and we obtain an nsd-k-coloring of G, a contradiction. (b) If the assertion is false, that means dH (ui ) = 2 for 1 ≤ i ≤ d. According to Claim 3.1(ii), dG (u1 ) = · · · = dG (ud ) = 2. Case 1: dG (u) = dH (u) = d. Let {vi } = N (ui )\{u} (1 ≤ i ≤ d). Then G must contain a subgraph isomorphic to the configuration F4 in Fig. 2. Consider G′ = G − {uui | 1 ≤ i ≤ d}, then G′ admits a good coloring c. Let Si be the set of available colors for each uui (1 ≤ i ≤ d). Observe that for 1 ≤ i ≤ d, |Si | = |[k]\({c (ui vi )} ∪ {w(vi ) − c (ui vi )})| ≥ ∆ + 2 − 2 = ∆ > 2d − 2. We associate with each uui a variable xi . Now we consider the following polynomial:



P (x1 , x2 , . . . , xd ) =

( xi − xj )

1≤i
 d d   l =1

 xk − xl − cl

,

(1)

k=1

where cl = c (ul vl ) (1 ≤ l ≤ d). By easy computations, cP (x12d−2 xd2−1 xd3−3 xd4−4 . . . xd−1 ) = 1 ̸= 0. According to Lemma 2.3, we can choose xi ∈ Si (1 ≤ i ≤ d) such that P (x1 , x2 , . . . , xd ) ̸= 0. So we get an nsd-k-coloring of G, a contradiction. Case 2: dG (u) > d. Let {vi } = N (ui )\{u} (1 ≤ i ≤ d). Denote by z1 , z2 , . . . , zt (1 ≤ t ≤ ∆ − d) the leaves adjacent to u in G, see F5 in Fig. 2. Let G′ = G − {zj | 1 ≤ j ≤ t }, then G′ admits a good coloring c. If t = 1, then we have to exclude the colors in {c (uui ) | 1 ≤ i ≤ d} ∪1≤j≤d {w(uj ) − w(u)} from those available for uz1 . ⌉ ≥ 1. Thus we obtain an nsd-k-coloring of So the number of available colors for uz1 is at least ∆ + 2 − 2d ≥ ∆ + 2 − 2⌈ ∆ 2 G, a contradiction. If 2 ≤ t ≤ ∆ − d, then the colors in {c (uui ) | 1 ≤ i ≤ d} are forbidden for uzj (1 ≤ j ≤ t ). Thus there exist ∆ + 2 − d available colors for each uzj (1 ≤ j ≤ t ). According to Lemma 2.2, we can color uzj (1 ≤ j ≤ t ) properly so that wG (u) ̸= wG (ui ) (1 ≤ i ≤ d) since t (∆ + 2 − d) − t 2 + 1 ≥ d + 1. So we get an nsd-k-coloring of G, a contradiction. (iv) If u is adjacent to a bad 2-vertex u1 , suppose that u1 , u2 , . . . , ud−2 are (d − 2) 2-vertices adjacent to u in the graph H. According to Claim 3.1(ii) and (iii), dG (u) = dH (u) = d, dG (ui ) = 2 (1 ≤ i ≤ d − 2). Set {vi } = N (ui )\{u} (1 ≤ i ≤ d − 2). Without loss of generality, we assume that dH (v1 ) = 3. Denote {y1 , y2 } = N (v1 )\{u1 }, see F6 in Fig. 2. Consider G′ = G−{uui | 1 ≤ i ≤ d − 2}, then G′ admits a good coloring c. Let Si be the set of available colors for each uui (1 ≤ i ≤ d − 2). Notice that the colors in {c (uud−1 ), c (uud )} ∪ {w(v1 ) − c (u1 v1 )} are forbidden for uu1 , and the colors in {c (uud−1 ), c (uud )} ∪ {c (ui vi )} ∪ {w(vi ) − c (ui vi )} are forbidden for uui (2 ≤ i ≤ d − 2). Observe that for 2 ≤ i ≤ d − 2, |Si | ≥ ∆ + 2 − 4 = ∆ − 2 > d − 3, |S1 | ≥ ∆ + 2 − 3 = ∆ − 1 > d − 2. We associate with each uui a variable xi (1 ≤ i ≤ d − 2). Then we consider the following polynomial:

 P (x1 , x2 , . . . , xd−2 ) =

 1≤i
(xi − xj )

d−2  k=1

  d−2  xk + cd−1 + cd − w(ud−1 ) xl + cd−1 + cd − w(ud ) , l =1

(2)

X. Yu et al. / Discrete Mathematics 339 (2016) 62–71

67

where cj = c (uuj ) for j = d − 1, d. Let

 Q (x1 , x2 , . . . , xd−2 ) =



( xi − xj )

1≤i
d−2 

2 xk

.

(3)

k=1

By Lemma 2.4, cP (xd1−2 xd2−3 xd3−5 xd4−6 . . . xd−3 ) = cQ (xd1−2 xd2−3 xd3−5 xd4−6 . . . xd−3 ) = 1 ̸= 0. By Lemma 2.3, there exist x1 ∈ S1 , . . . , xd−2 ∈ Sd−2 such that P (x1 , x2 , . . . , xd−2 ) ̸= 0. We color uui (1 ≤ i ≤ d − 2) correspondingly. Then we recolor u1 v1 with a color α ̸∈ {c (uu1 )} ∪ {c (v1 y1 ), c (v1 y2 )} ∪ {w(y1 ) − (w(v1 ) − c (u1 v1 ))} ∪ {w(y2 ) − (w(v1 ) − c (u1 v1 ))}. ⌉ > ∆ + 2 that wG (u) ̸= wG (ui ) (1 ≤ i ≤ d − 2). So we get an nsd-k-coloring It follows from the inequality 1 + 2 + · · · + ⌈ ∆ 2 of G, a contradiction. This completes the proof of Claim 3.2(iv).  In order to complete the proof, we use a discharging technique on the vertices of the graph H by defining the weight function w : V (H ) → R with w(v) = dH (v) for v ∈ V (H ). It follows from the hypothesis on the maximum average degree (mad(H ) < 3) that the total sum of weights is strictly less than 3|V (H )|. Then we define discharging rules to redistribute weights of the vertices in the graph H, and once the discharging is finished, a new weight function w ∗ will be produced such that during the discharging process, the total sum of weights will be kept fixed. This will lead to the following contradiction: 3|V (H )| ≤



w∗ (v) =

v∈V (H )



w(v) < 3|V (H )|

v∈V (H )

and hence this counterexample cannot exist. The discharging rules are defined as follows: (R1) Every (⌈ ∆ ⌉ + 1)+ -vertex gives 1 to each adjacent bad 2-vertex. 2 (R2) Every 4+ -vertex gives 12 to each adjacent good 2-vertex. Let v ∈ V (H ) be a d-vertex. By Claim 3.1(i), d ≥ 2. Consider the following cases: Case 1: d = 2. Observe that w(v) = 2. According to Claim 3.2(i), v is adjacent to at least one 6+ -vertex. If v is a good 2-vertex, by (R2), w ∗ (v) = 2 + 2 × 12 = 3. If v is a bad 2-vertex, by (R1) and Claim 3.2(iii) (a), w ∗ (v) = 2 + 1 × 1 = 3. Case 2: d = 3. Observe that w ∗ (v) = w(v) = 3. Case 3: d = 4. Observe that w(v) = 4. By Claim 3.2(i) and (ii), v is adjacent to at most one good 2-vertex and is not adjacent to any bad 2-vertex. By (R2), w ∗ (v) ≥ 4 − 1 × 12 > 3. Case 4: 5 ≤ d ≤ ⌈ ∆ ⌉. Observe that w(v) = d. By Claim 3.2(iii), v is adjacent to at most (d − 1) good 2-vertices and is not 2

= 12 (d + 1) ≥ 3. Case 5: d ≥ ⌈ ∆ ⌉ + 1. Observe that w(v) = d. By Claim 3.2(iv), if v is adjacent to a bad 2-vertex, then v is adjacent to at 2 most (d − 3) 2-vertices. By (R1) and (R2), w ∗ (v) ≥ d − (d − 3) × 1 = 3. Otherwise, v is adjacent to at most d good 2-vertices. By (R2), w ∗ (v) ≥ d − d × 12 = 21 d ≥ 3. adjacent to any bad 2-vertex. By (R2), w ∗ (v) ≥ d − (d − 1) ×

1 2

This completes the proof. 3.2. Case 5 ≤ ∆ ≤ 8 Again, we give some structural properties of the graph H. Claim 3.3. The graph H has the following properties: (i) It holds that δ(H ) ≥ 2. (ii) If u ∈ V (H ) and dH (u) ≤ 3, then dG (u) = dH (u). (iii) If uv r is a path in H such that dH (v) = 2 and 2 ≤ dH (r ) ≤ 3, then dG (u) = dH (u). The proof of Claim 3.3 is analogous to that in Claim 3.1 and is left to the reader. Claim 3.4. Let u ∈ V (H ), dH (u) = d, uui ∈ E (H ), 1 ≤ i ≤ d. (i) If d = 2, then u is adjacent to at most one 3− -vertex. (ii) Suppose that d = 4. (a) If u is adjacent to a bad 2-vertex, then the other neighbors of u in the graph H are 3+ -vertices. (b) If u is not adjacent to any bad 2-vertex, then u is adjacent to at most two good 2-vertices. (iii) Suppose that d = 5. (a) If u is adjacent to two bad 2-vertices, then the other neighbors of u in the graph H are 3+ -vertices. (b) If u is adjacent to exactly one bad 2-vertex, then u is adjacent to at most two good 2-vertices. (c) If u is not adjacent to any bad 2-vertex, then u is adjacent to at most four good 2-vertices. (iv) If d ≥ 6 and u is adjacent to a bad 2-vertex, then u is adjacent to at most (d − 3) 2-vertices.

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Fig. 3. Illustration of Claim 3.4.

Proof. Since each polynomial P in this subsection is specific, we can use MATLAB to compute the coefficient of the corresponding monomial in each polynomial P which we need. (i) The proof is similar to the proof of Claim 3.2(i), so we omit it. (ii) (a) If u is adjacent to a bad 2-vertex u1 , suppose that dH (u1 ) = dH (u2 ) = 2. Let {vi } = N (ui )\{u} (i = 1, 2), and let N (v1 ) = {u1 , y1 , y2 }. Without loss of generality, we may assume that dH (v1 ) = 3. By Claim 3.3(ii) and (iii), dG (u1 ) = dG (u2 ) = 2, dG (v1 ) = dH (v1 ) = 3, dG (u) = dH (u) = 4, see F1 in Fig. 3. Consider G′ = G − {uu1 , uu2 }, then G′ admits a good coloring c. Let Si be the set of available colors for each uui (i = 1, 2). Notice that the colors in {c (uu3 ), c (uu4 )} ∪ {w(v1 ) − c (u1 v1 )} ∪ {c (u2 v2 ) − (c (uu3 ) + c (uu4 ))} are forbidden for uu1 , and the colors in {c (uu3 ), c (uu4 )} ∪ {c (u2 v2 )} ∪ {w(v2 ) − c (u2 v2 )} are forbidden for uu2 . It is easy to see that |Si | ≥ 7 − 4 = 3 (i = 1, 2). By Lemma 2.1, we can choose αi ∈ Si (i = 1, 2) such that α1 ̸= α2 and wG (u) ̸= wG (uj ) for j = 3, 4. Recolor u1 v1 with a color α ̸∈ {c (uu1 )} ∪ {c (v1 y1 ), c (v1 y2 )} ∪

{

4

i=2

c (uui )} ∪j=1,2 {w(yj ) − (w(v1 ) − c (u1 v1 ))}. Thus we get an nsd-k-coloring of G, which is a contradiction.

(b) First, we assume that dG (u) = 4. If the assertion is false, denote by u1 , u2 , u3 the three good 2-vertices adjacent to u in the graph H. Put N (ui ) = {u, vi } (1 ≤ i ≤ 3). According to Claim 3.3(ii), dG (ui ) = 2 (1 ≤ i ≤ 3), see F2 in Fig. 3. Consider G′ = G − {uu1 , uu2 , uu3 }, then G′ admits a good coloring c. Case 1: 6 ≤ ∆(G) ≤ 8. Because the colors in {c (uu4 )} ∪ {c (ui vi )} ∪ {w(vi ) − c (ui vi )} are forbidden for uui (1 ≤ i ≤ 3), the number of available colors for uui (1 ≤ i ≤ 3) is at least ∆ + 2 − 3 ≥ 5. Now we associate with each uui a variable xi (1 ≤ i ≤ 3). Then we consider the following polynomial: P (x1 , x2 , x3 ) =



(xi − xj )(x2 + x3 + c4 − c1 )(x1 + x3 + c4 − c2 )

1≤i
× (x1 + x2 + c4 − c3 )(x1 + x2 + x3 + c4 − w(u4 )),

(4)

where ci = c (ui vi ) (1 ≤ i ≤ 3) and c4 = c (uu4 ). By computer computations, cP (x41 x32 ) = 1. According to Lemma 2.3, there exists xi ∈ Si , 1 ≤ i ≤ 3 such that P (x1 , x2 , x3 ) ̸= 0. We color uui (1 ≤ i ≤ 3) correspondingly. So we can extend the coloring c to the whole graph G, which is a contradiction. Case 2: ∆(G) = 5. If w(vi ) − c (ui vi ) ̸∈ {6, 7} for some i ∈ {1, 2, 3}, from the fact that vi is a 4+ -vertex, it follows that w(vi ) − c (ui vi ) > 7 and the sum at ui will be always different from the sum at vi . It is easy to see that the colors in {c (uu4 )} ∪ {c (ui vi )} are forbidden for uui and the colors in {c (uu4 )} ∪ {c (ul vl )} ∪ {w(vl ) − c (ul vl )} are forbidden for uul (l ∈ [3]\{i}). So we have at least ∆ + 2 − 2 = 5 available colors for uui and ∆ + 2 − 3 = 4 available colors for uul (l ∈ [3]\{i}). Without loss of generality, we assume that i = 1. Using the same argument as in the proof of Case 1 above, we can easily extend the coloring c to the whole graph G, which is a contradiction. Otherwise, that means w(vi ) − c (ui vi ) ∈ {6, 7} for 1 ≤ i ≤ 3. Now we consider the following subcases: Subcase 2.1: c (ul vl ) ̸∈ {6, 7} for some l ∈ {1, 2, 3}, without loss of generality, we assume that l = 1. Obviously, the colors in {c (uu4 )}∪{c (ui vi )}∪{w(vi )− c (ui vi )} are forbidden for uui (1 ≤ i ≤ 3). So we have at least ∆ + 2 − 3 = 4 available colors

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69

for each uui (1 ≤ i ≤ 3). We associate with each uui a variable xi (1 ≤ i ≤ 3). Then we consider the following polynomial: P (x1 , x2 , x3 ) =



(xi − xj )(x1 + x3 + c4 − c2 )

1≤i
× (x1 + x2 + c4 − c3 )(x1 + x2 + x3 + c4 − w(u4 )),

(5)

where ci = c (ui vi ) for i = 2, 3 and c4 = c (uu4 ). By computer computations, ( ) = −1. According to Lemma 2.3, there exists xi ∈ Si (1 ≤ i ≤ 3) such that P (x1 , x2 , x3 ) ̸= 0. We color uui (1 ≤ i ≤ 3) correspondingly. Because wG (u) ̸= wG (u1 ), we can extend the coloring c to the whole graph G, a contradiction. Subcase 2.2: c (ui vi ) ∈ {6, 7} for every 1 ≤ i ≤ 3. It is easy to see that the colors in {c (uu4 )} ∪ {c (ui vi )} ∪ {w(vi ) − c (ui vi )} are forbidden for uui (1 ≤ i ≤ 3). In the following we extend the  coloring c to the whole graph G. Denote by Si the set of available colors for each uui , where 1 ≤ i ≤ 3. Put α = max( 1≤i≤3 Si ). We choose Sl (l ∈ {1, 2, 3}) with the minimum  subscript such that α ∈ Sl , and color uul with α . Set β = max( j∈([3]\{l}) Sj \{α}). Then we choose Sj with the minimum  subscript such that β ∈ Sj , and color uuj with β . Similarly, let γ = max( k∈([3]\{l,j}) Sk \{α, β}) and we color uuk with γ . Now we observe that wG (u) ̸= wG (ui ) for 1 ≤ i ≤ 3. If wG (u) ̸= wG (u4 ), we are done. Otherwise, we can recolor the edge taking the color β or γ so that wG (u) ̸= wG (u4 ) and we are done. In the following we assume that dG (u) > 4. Denote by z1 , z2 , . . . , zt (1 ≤ t ≤ ∆ − 4) the leaves adjacent to u in G. According to Claim 3.3(ii), dG (ui ) = dH (ui ) = 2 for 1 ≤ i ≤ 3, see F3 in Fig. 3. Consider G′ = G − {zj | 1 ≤ j ≤ t }, then G′ admits a good coloring c. The colors in {c (uui ) | 1 ≤ i ≤ 4} are forbidden for uzj (1 ≤ j ≤ t ). Case 1: t = 1. When 5 ≤ ∆ ≤ 7, notice that the color w(u4 ) − c (uu4 ) is also forbidden for uz1 , thus we have at least ∆ + 2 − 4 − 1 = ∆ − 3 available colors for uz1 . It follows from the inequality 1 + 2 + 3 + 4 > ∆ + 2 that wG (u) ̸= wG (ui ) (1 ≤ i ≤ 3). So we can obtain an nsd-k-coloring of G, which is a contradiction. When 8 ≤ ∆ ≤ 9, observe that the colors in {w(ui ) − c (uui )} (1 ≤ i ≤ 4) are also forbidden for uz1 , so there exist at least ∆ + 2 − 4 − 4 = ∆ − 6 available colors for uz1 , a contradiction. Case 2: 2 ≤ t ≤ ∆ − 4. Notice that there exist at least ∆ + 2 − 4 = ∆ − 2 available colors for each uzj (1 ≤ j ≤ t ). According to Lemma 2.2, we can color uzj (1 ≤ j ≤ t ) properly so that wG (u) ̸= wG (u4 ). Since 1 + 2 + · · · + 5 > ∆ + 2, it follows that wG (u) ̸= wG (ui ) (1 ≤ i ≤ 3). Thus we may extend the coloring c to the whole graph G, a contradiction. (iii) (a) When u is adjacent to two bad 2-vertices u1 and u2 , suppose that there exists a 2-vertex u3 adjacent to u in H. Denote N (ui ) = {u, vi }, where 1 ≤ i ≤ 3. Without loss of generality, we may assume that dH (vi ) = 3 (i = 1, 2). Set N (vi ) = {ui , yi1 , yi2 }, where i = 1, 2. According to Claim 3.3(ii) and (iii), dG (u) = dH (u) = 5, dG (ui ) = 2 (1 ≤ i ≤ 3), dG (vi ) = dH (vi ) = 3 (i = 1, 2), see F4 in Fig. 3. Consider G′ = G − {uu1 , uu2 , uu3 }, then G′ admits a good coloring c. Let Si be the set of available colors for each uui (1 ≤ i ≤ 3). Now we associate with each uui a variable xi (1 ≤ i ≤ 3). Then we consider the following polynomial: cP x31 x32

P (x1 , x2 , x3 ) =



(xi − xj )(x1 + x2 + c4 + c5 − c3 )(x1 + x2 + x3 + c4 + c5 − w(u4 ))

1≤i
× (x1 + x2 + x3 + c4 + c5 − w(u5 )),

(6)

where ci = c (uui ) for i = 4, 5 and c3 = c (u3 v3 ). It is easy to see that the colors in {c (uu4 ), c (uu5 )} ∪ {w(vi ) − c (ui vi )} are forbidden for uui (i = 1, 2), the colors in {c (uu4 ), c (uu5 )} ∪ {c (u3 v3 )} ∪ {w(v3 ) − c (u3 v3 )} are forbidden for uu3 . Thus we have that |Si | ≥ ∆ + 2 − 3 ≥ 4 (i = 1, 2), |S3 | ≥ ∆ + 2 − 4 ≥ 3. By computer computations, cP (x31 x2 x23 ) = −1. According to Lemma 2.3, there exists xi ∈ Si (1 ≤ i ≤ 3) such that P (x1 , x2 , x3 ) ̸= 0. We color uui (1 ≤ i ≤ 3) correspondingly. Then 5 we can recolor ui vi with a color αi ̸∈ {c (uui )} ∪ { l=3 c (uul ) + c (uu3−i )} ∪ {c (vi yi1 ), c (vi yi2 )} ∪1≤j≤2 {w(yij ) − (w(vi ) − c (ui vi ))} (1 ≤ i ≤ 2). So we obtain an nsd-k-coloring of G, which is a contradiction. (b) When u is adjacent to exactly one bad 2-vertex u1 , assume that u2 , u3 , u4 are three good 2-vertices adjacent to u in H. Set N (ui ) = {u, vi } where 1 ≤ i ≤ 4. Without loss of generality, we may assume that dH (v1 ) = 3. Let N (v1 ) = {u1 , y11 , y12 }. By Claim 3.3(ii) and (iii), dG (u) = dH (u) = 5, dG (ui ) = 2 (1 ≤ i ≤ 4), dG (v1 ) = dH (v1 ) = 3, see F5 in Fig. 3. Let G′ = G−{uu1 , uu2 , uu3 , uu4 }, then G′ admits a good coloring c. Put Si as the set of available colors for each uui (1 ≤ i ≤ 4). Observe that the colors in {c (uu5 )}∪{w(v1 )− c (u1 v1 )} are forbidden for uu1 , the colors in {c (uu5 )}∪{c (ui vi )}∪{w(vi )− c (ui vi )} are forbidden for uui , where 2 ≤ i ≤ 4. So |S1 | ≥ ∆ + 2 − 2 = ∆, |Si | ≥ ∆ + 2 − 3 = ∆ − 1 (2 ≤ i ≤ 4). Now we associate with each uui a variable xi (1 ≤ i ≤ 4). If ∆ ≥ 6, we consider the following polynomial: P (x1 , x2 , x3 , x4 ) =



(xi − xj )(x1 + x3 + x4 + c5 − c2 )(x1 + x2 + x4 + c5 − c3 )

1≤i
× (x1 + x2 + x3 + c5 − c4 )(x1 + x2 + x3 + x4 + c5 − w(u5 )),

(7)

where ci = c (ui vi ) for 2 ≤ i ≤ 4 and c5 = c (uu5 ). By computer computations, cP (x51 x33 x24 ) = 2. According to Lemma 2.3, there exists xi ∈ Si (1 ≤ i ≤ 4) such that P (x1 , . . . , x4 ) ̸= 0. We color uui (1 ≤ i ≤ 4) correspondingly. Now we can recolor u1 v1 5 with a color α ̸∈ {c (uu1 )}∪{ i=2 c (uui )}∪{c (v1 y11 ), c (v1 y12 )}∪{w(y11 )−(w(v1 )−c (u1 v1 ))}∪{w(y12 )−(w(v1 )−c (u1 v1 ))}. So we obtain an nsd-k-coloring of G, a contradiction.

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If ∆ = 5, we consider the following polynomial:



P (x1 , x2 , x3 , x4 ) =

(xi − xj )(x1 + x2 + x3 + x4 + c5 − w(u5 )),

(8)

1≤i
where c5 = c (uu5 ). By computer computations, cP (x41 x22 x3 ) = 1. According to Lemma 2.3, there exists xi ∈ Si (1 ≤ i ≤ 4) such that P (x1 , . . . , x4 ) ̸= 0. We color uui (1 ≤ i ≤ 4) correspondingly. Because 1 + 2 + 3 + 4 + ci > ci + 7, where ci = c (uui ) for 1 ≤ i ≤ 4, it follows that wG (u) ̸= wG (ui ) (1 ≤ i ≤ 4). Now we recolor u1 v1 as in the case ∆ ≥ 6 above and obtain an nsd-k-coloring of G. This completes the proof of Claim 3.4(iii)(b). (c) If u is not adjacent to any bad 2-vertex, suppose that u is adjacent to five good 2-vertices. By Claim 3.3(ii), dG (ui ) = 2 (1 ≤ i ≤ 5). Case 1: dG (u) = dH (u) = 5. Set N (ui ) = {u, vi }, where 1 ≤ i ≤ 5, see F6 in Fig. 3. Let G′ = G − {uui | 1 ≤ i ≤ 5}, then G′ admits a good coloring c. Apparently, the colors in {c (ui vi )}∪{w(vi )− c (ui vi )} are forbidden for uui for 1 ≤ i ≤ 5. So we have at least ∆ + 2 − 2 = ∆ available colors for each uui (1 ≤ i ≤ 5). Now we associate with each uui a variable xi (1 ≤ i ≤ 5). If ∆ ≥ 7, we consider the following polynomial: P ( x1 , x2 , . . . , x5 ) =



( xi − xj )

1≤i
 5 5   l =1

 xk − xl − cl

,

(9)

k=1

where cl = c (ul vl ) (1 ≤ l ≤ 5). By computer computations, cP (x42 x33 x64 x25 ) = 2. By Lemma 2.3, it is sufficient to prove that P (x1 , x2 , . . . , x5 ) ̸= 0 for some colors x1 , . . . , x5 available for the corresponding edges. So we obtain an nsd-k-coloring of G, which is a contradiction. If 5 ≤ ∆ ≤ 6, we can color each uui (1 ≤ i ≤ 5) properly. Because 1 + 2 + 3 + 4 + ci > ci + 8, where ci = c (uui ) for 1 ≤ i ≤ 5, it follows that wG (u) ̸= wG (ui ) (1 ≤ i ≤ 5). Finally, we obtain an nsd-k-coloring of G, a contradiction. Case 2: dG (u) > dH (u) = 5. Denote by z1 , z2 , . . . , zt (1 ≤ t ≤ ∆ − 5) the leaves adjacent to u in the graph G, see F7 in Fig. 3. Let G′ = G − {zj | 1 ≤ j ≤ t }, then G′ admits a good coloring c. The colors in {c (uui ) | 1 ≤ i ≤ 5} are forbidden for each uzj (1 ≤ j ≤ t ). So we have at least ∆ + 2 − 5 = ∆ − 3 available colors for each uzj (1 ≤ j ≤ t ) and we can color each uzj (1 ≤ j ≤ t ) properly. From the fact that 1 + 2 + 3 + 4 + 5 + ci > ci + 10, where ci = c (uui ) for 1 ≤ i ≤ 5, it follows that wG (u) ̸= wG (ui ) (1 ≤ i ≤ 5). Finally, we get an nsd-k-coloring of G, a contradiction. (iv) From the fact that 1 + 2 + 3 + 4 + 5 + ci > ci + 10, where ci = c (uui ) for 1 ≤ i ≤ 5, it follows that wG (u) ̸= wG (ui ) (1 ≤ i ≤ 5). The remaining proof of the argument is similar to that in Claim 3.2(iv) and will not be reproduced here.  Again, we define an initial weight function w : V (H ) → R with w(v) = dH (v) for every vertex v ∈ V (H ), and set the following discharging rules: (R1) Every 4+ -vertex gives 1 to each adjacent bad 2-vertex. (R2) Every 4+ -vertex gives 21 to each adjacent good 2-vertex. Denote by w ∗ (v) the new charge function after the discharging process is complete. It is sufficient to show that w ∗ (v) ≥ 3 for all v ∈ V (H ), what finally will lead to the following contradiction: 3|V (H )| ≤

 v∈V (H )

w ∗ (v) =



w(v) < 3|V (H )|.

v∈V (H )

Let v ∈ V (H ) be a d-vertex. By Claim 3.3(i), d ≥ 2. Consider the following cases: Case 1: d = 2. Observe that w(v) = 2. According to Claim 3.4(i), v is adjacent to at least one 4+ -vertex. If v is a good 2-vertex, by (R2), w ∗ (v) = 2 + 2 × 21 = 3. If v is a bad 2-vertex, by (R1), w ∗ (v) = 2 + 1 × 1 = 3. Case 2: d = 3. Observe that w ∗ (v) = w(v) = 3. Case 3: d = 4. Observe that w(v) = 4. According to Claim 3.4(ii), v might be adjacent to one bad 2-vertex (and to no other 2-vertex). Otherwise, v is adjacent to at most two good 2-vertices (and to no other 2-vertex). If v is adjacent to one bad 2-vertex, by (R1), w ∗ (v) ≥ 4 − 1 × 1 = 3. Otherwise, by (R2), w ∗ (v) ≥ 4 − 2 × 12 = 3. Case 4: d = 5. Observe that w(v) = 5. By Claim 3.4(iii) (a), v is adjacent to at most two bad 2-vertices. If v is adjacent to exactly two bad 2-vertices, by Claim 3.4(iii)(a) and (R1), w ∗ (v) = 5 − 2 × 1 = 3. If v is adjacent to exactly one bad 2-vertex, by Claim 3.4(iii)(b), (R1) and (R2), w ∗ (v) ≥ 5 − 1 × 1 − 2 × 12 = 3. If v is not adjacent to any bad 2-vertex, according to Claim 3.4(iii)(c), v is adjacent to at most four good 2-vertices. By (R2), w ∗ (v) ≥ 5 − 4 × 12 = 3. Case 5: d ≥ 6. Observe that w(v) = d. By Claim 3.4(iv), if v is adjacent to a bad 2-vertex, then v is adjacent to at most (d − 3) 2-vertices. By (R1) and (R2), w ∗ (v) ≥ d − (d − 3) × 1 = 3. Otherwise, v is adjacent to at most d good 2-vertices. By (R2), w ∗ (v) ≥ d − d × 21 ≥ 3. This completes the proof. Acknowledgments This work was supported by the National Natural Science Foundation of China (11101243, 11471193), the Scientific Research Foundation for the Excellent Middle-Aged and Young Scientists of Shandong Province of China (BS2012SF016),

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the Fundamental Research Funds of Shandong University and Independent Innovation Foundation of Shandong University (IFYT14012). We also appreciate the referees’ great efforts on our paper for their valuable suggestions and comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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