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Edge-partitions of graphs and their neighbor-distinguishing index
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Edge-partitions of graphs and their neighbor-distinguishing index Bojan Vučković Mathematical Institute, Serbian Academy of Science and Arts, Kneza Mihaila 36 (P.O. Box 367), 11001 Belgrade, Serbia
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Article history: Received 1 March 2017 Received in revised form 4 July 2017 Accepted 5 July 2017 Available online xxxx Keywords: Neighbor-distinguishing edge coloring Maximum degree Edge-partition
a b s t r a c t A proper edge coloring is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The minimum number of colors needed for a neighbor-distinguishing edge coloring is the neighbor-distinguishing index, denoted by χa′ (G). A graph is normal if it contains no isolated edges. Let G be a normal graph, and let ∆(G) and χ ′ (G) denote the maximum degree and the chromatic index of G, respectively. We modify the previously known techniques of edge-partitioning to prove that χa′ (G) ≤ 2χ ′ (G), which implies that χa′ (G) ≤ 2∆(G) + 2. This improves the result in Wang et al. (2015), which states that χa′ (G) ≤ 25 ∆(G) for any normal graph. We also prove that χa′ (G) ≤ 2∆(G) when ∆(G) = 2k , k is an integer with k ≥ 2. © 2017 Elsevier B.V. All rights reserved.
1. Introduction All graphs considered in this paper are simple and finite. Let V (G), E(G), and ∆(G) denote the vertex set, the edge set, and the maximum degree of a graph G, respectively. Let NG (v ) and degG (v ) = |NG (v )| denote the set of neighbors and the degree of a vertex ⋃mv in G, respectively. An edge-partition of a graph G into subgraphs G1 , . . . , Gm is a decomposition of G such that E(G) = i=1 E(Gi ) and E(Gi ) ∩ E(Gj ) = ∅ for any pair i ̸ = j. For a graph G and any S ⊆ E(G), the edge-induced subgraph G[S ] is the subgraph of G whose edge set is S and whose vertex set consists of all end vertices of the edges in S. A proper edge k-coloring of a graph G is a function φ : E(G) → {1, . . . , k} such that every two adjacent edges receive different colors. The chromatic index χ ′ (G) of a graph G is the minimum positive integer k for which G has a proper edge k-coloring. Given an edge k-coloring φ of G, we use Cφ (v ) to denote the set of colors assigned to the edges incident with v . The edge coloring φ is called neighbor-distinguishing (in some papers adjacent vertex distinguishing), or nde-coloring for short, if Cφ (u) ̸ = Cφ (v ) for any pair of adjacent vertices u and v . The neighbor-distinguishing index χa′ (G) of a graph G is the smallest integer k such that G has a k-nde-coloring. A graph G is normal if it contains no isolated edges. It is obvious that G has an nde-coloring if and only if G is normal; thus we consider only normal graphs when examining an nde-coloring. Zhang, Liu and Wang [7] introduced and investigated a neighbor-distinguishing edge coloring of graphs, where they proposed the following conjecture. Conjecture 1. If G is a connected normal graph different from a 5-cycle, then χa′ (G) ≤ ∆(G) + 2. Akbari, Bidkhori and Nosrati [1] proved that χa′ (G) ≤ 3∆(G) for any normal graph G. Zhang, Wang and Lih [6] proved a better upper bound, χa′ (G) ≤ 25 ∆(G) + 5. Wang, Wang and Huo [5] improved this result by showing that χa′ (G) ≤ 25 ∆(G). Our main result is the following improvement of the upper bound for χa′ . E-mail address: [email protected]. http://dx.doi.org/10.1016/j.disc.2017.07.005 0012-365X/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: B. Vučković, Edge-partitions of graphs and their neighbor-distinguishing index, Discrete Mathematics (2017), http://dx.doi.org/10.1016/j.disc.2017.07.005.
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Theorem 2. If G is a normal graph, then χa′ (G) ≤ 2χ ′ (G). According to the famous Vizing’s theorem [4], χ ′ (G) ≤ ∆(G)+1. Thus as an immediate consequence we have the following corollary. Corollary 3. If G is a normal graph, then χa′ (G) ≤ 2∆(G) + 2. Furthermore, χa′ (G) of a normal graph G with ∆(G) = 2k , k ∈ N, k > 2, does not exceed 2∆(G), as stated in the next theorem. Theorem 4. If G is a normal graph with ∆(G) ≤ 2k , where k is an integer, and k ≥ 2, then χa′ (G) ≤ 2k+1 . The proofs of Theorems 2 and 4 are deferred to Section 3. First, we prove the statements that we will use in the proofs of these two theorems. 2. Preliminaries Ballister et al. [2] proved the first part, while Wang et al. [5] proved the second and third parts of the following theorem. Theorem 5. Let G be a normal graph. 1. If ∆(G) ≤ 3, then χa′ (G) ≤ 5. 2. If ∆(G) ≤ 4, then χa′ (G) ≤ 8. 3. If ∆(G) ≤ 5, then χa′ (G) ≤ 10. The following lemma and theorem, proved in [6], are the main tools used in papers [6] and [5] to attain the upper bounds. Lemma 6. If normal graph G has an edge-partition into normal graphs G1 and G2 , then χa′ (G) ≤ χa′ (G1 ) + χa′ (G2 ). Theorem 7. Let G be a normal graph with ∆(G) ≥ 6. Then there is an edge-partition of G into normal graphs H1 and H2 , such that: 1. ∆(H1 ) ≤ 3, 2. ∆(H2 ) ≤ ∆(G) − 2. The bound χa′ (G) ≤ 52 ∆(G) was proved by repeatedly applying the theorem above, and relying on the bounds of χa′ (H) for graphs with ∆(H) ≤ 5. We propose a procedure that is also edge-partitioning of a graph into two normal graphs, and then make use of Lemma 6. The difference is that we show how to produce an edge-partition of a normal graph G that has χ ′ (G) = k into normal graphs H1 and H2 with χa′ (H1 ) ≤ 4 and χ ′ (H2 ) ≤ k − 2. Consequently, using the bounds from Theorem 5 we get that χa′ (G) ≤ 2χ ′ (G) for any normal graph G. Define a function wG : E(G) → N by wG (uv ) = deg(u) + deg(v ) for each uv ∈ E(G). We use the next theorem in the proofs of Theorems 2 and 4 to show that a minimal counterexample of a graph does not contain an edge uv with wG (uv ) ≤ ∆(G) + 2. Theorem 8. Assume that there exists a connected graph G with ∆(G) ≥ 4 and χa′ (G) > 2k where ∆(G) ≤ k. If wG (uv ) ≤ ∆(G) + 2 for some uv ∈ E(G), then there exists a normal graph H with ∆(H) ≤ ∆(G), |E(H)| < |E(G)|, and χa′ (H) > 2k. Proof. Suppose that the statement is false. Let uv be an edge with wG (uv ) ≤ ∆(G) + 2, and let H = G − uv . Clearly, ∆(H) ≤ ∆(G) and |E(H)| < |E(G)|. Thus if H is a normal graph, then χa′ (H) ≤ 2k. Denote by L the set of colors {1, . . . , 2k}. Assume first that H is not a normal graph. Since G is a normal graph, this means that one of the vertices u and v , say v , has only one adjacent vertex w in H, and degH (w ) = 1. Now let H ′ = G − vw . Then degH ′ (v ) = 1 and degH ′ (w ) = 0. Also, degH ′ (u) > 1 since G is a connected graph with ∆(G) ≥ 4. Thus H ′ is a normal graph. Then there exists an nde-coloring σ of H ′ with colors from L. Let φ be an edge coloring of G with φ (e) = σ (e) for every e ∈ E(H ′ ). Assign to vw any color from L different from φ (uv ), and if degG (u) = degG (v ) = 2, a color not in Cφ (u). Since |L| > 2, this can always be done; thus φ is an nde-coloring with not more than 2k colors. This produces a contradiction. Therefore, we may assume that H is a normal graph, implying that χa′ (H) ≤ 2k. Depending on whether vertices u and v have a common neighbor, we consider two cases. 1. NG (u) ∩ NG (v ) = ∅. Let H ′ be a graph obtained from G by contracting the edge uv , and denote by w the vertex obtained by identifying u and v . Since degH ′ (w ) = degG (u) + degG (v ) − 2 ≤ ∆(G), it follows that ∆(H ′ ) ≤ ∆(G). Hence |E(H ′ )| < |E(G)| and so χa′ (H ′ ) ≤ 2k. Let σ ′ be an nde-coloring of H ′ with colors from L. Next, define an edge coloring σ of H in the following manner: Please cite this article in press as: B. Vučković, Edge-partitions of graphs and their neighbor-distinguishing index, Discrete Mathematics (2017), http://dx.doi.org/10.1016/j.disc.2017.07.005.
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• for every x ∈ NH (u), assign σ (ux) = σ ′ (wx), • for every y ∈ NH (v ), assign σ (v y) = σ ′ (wy), • to any other edge from H, assign the color assigned by σ ′ to the corresponding edge from H ′ . Edge coloring σ is proper, but not necessarily an nde-coloring of H. Since all the edges incident with w in H ′ are assigned different colors by σ ′ , we have Cσ (u) ∩ Cσ (v ) = ∅. Next, we define an nde-coloring φ of G with colors from L, which produces a contradiction. To each edge of G except uv assign the color assigned by σ . Thus we are left to color the edge uv . Suppose that wG (uv ) < k + 2. Then there are at most k − 1 colors assigned to the edges adjacent to uv . Also, there are at most k − 1 colors to avoid for uv to obtain that Cφ (u) is different from Cφ (u′ ) for every u′ ∈ NG (u) − v , and Cφ (v ) is different from Cφ (v ′ ) for every v ′ ∈ NG (v ) − u. Hence we color uv with any of the remaining colors from L. Since Cσ (u) ̸ = Cσ (v ), φ is an nde-coloring of G with 2k colors, which is a contradiction. Hence we assume that wG (uv ) = k + 2. Let d = degH (u) = degG (u) − 1, L1 = {1, . . . , d}, and L2 = {1, . . . , k − d}. Denote by ui , where i ∈ L1 , the vertices from NG (u) − v , and by vj , where j ∈ L2 , the vertices from NG (v ) − u. We may assume, without loss of generality, that φ (uui ) = i for every i ∈ L1 , φ (vvj ) = d + j for every j ∈ L2 , and d ≤ 2k . If there exists a color, say l, from K = {k + 1, . . . , 2k} such that Cσ (u) ∪ {l} ̸ = Cφ (ui ) for every i ∈ L1 and Cσ (v ) ∪ {l} ̸ = Cφ (vj ) for every j ∈ L2 , color uv with l. The coloring φ produced is an nde-coloring, which is a contradiction. Therefore, we may assume that Cφ (ui ) = {1, . . . , d} ∪ {k + i} for every i ∈ L1 , and Cφ (vj ) = {d + 1, . . . , k} ∪ {k + d + j} for every j ∈ L2 . Next, we change the color of uu1 , keeping the edge coloring proper, and Cφ (u1 ) ̸ = Cφ (x) for every x ∈ NG (u1 ) − u. We know that Cσ (u) ⊂ Cφ (u1 ) and |Cφ (u1 )| = d + 1. In addition, |NG (u1 ) − u| = d and d ≤ 2k . Thus to keep the edge coloring proper, and Cφ (u1 ) ̸ = Cφ (x) for every x ∈ NG (u1 ) − u, we need to avoid for uu1 at most 2d + 1 ≤ k + 1 colors. We change the color of uu1 to any of the remaining k − 1 colors. At last, assign the color 1 to uv . Such a coloring φ is an nde-coloring of G with 2k colors, which is a contradiction. 2. NG (u) ∩ NG (v ) ̸ = ∅. Denote by w one of the common neighbors of u and v in H. Let σ be an nde-coloring of H with colors from L. Next, let φ be an edge coloring of G that assigns the same color as σ to every edge of G − uv . Since wG (uv ) ≤ k + 2 and NH (u) ∩ NH (v ) ̸ = ∅, we have |NH (u) ∪ NH (v )| < k. Hence there are at most k − 1 vertices in H that are adjacent to u or v , or both. In addition, there are at most k edges adjacent to uv in G. Thus we can choose at least one of the colors from L for uv , call it l, so that the coloring is proper, and also Cφ (u) ̸ = Cφ (u′ ) for every u′ ∈ NG (u) − v , and Cφ (v ) ̸ = Cφ (v ′ ) for every v ′ ∈ NG (v ) − u. If Cσ (u) ̸ = Cσ (v ), assign l to uv , and we obtain an nde-coloring φ . In the described coloring, at most 2k colors were used, which contradicts the assumption that χa′ (G) > 2k. Therefore, we may assume that Cσ (u) = Cσ (v ), which implies that degH (u) = degH (v ) = s and k ≥ 2s. We now change the colors of uw and vw in φ while keeping the coloring proper and paying attention that Cφ (w ) remains different from Cφ (x) for every x ∈ NG (w ) \ {u, v}. Let n = |NG (w ) \ {u, v}|. Since ∆(G) ≤ k, we have n ≤ k − 2. Let L1 = {σ (uw )} ∪ L \ (Cσ (u) ∪ Cσ (w )) and L2 = {σ (vw )} ∪ L \ (Cσ (v ) ∪ Cσ (w )). We have that k
|L1 | ≥ 2k − (k − 2 + s) + 1 = k − s + 3 ≥ ⌈ ⌉ + 3 2
and |L2 | ≥ ⌈ 2k ⌉ + 3. Next, we choose a color from L1 for uw , and a color from L2 for vw , so that φ (uw ) ̸ = φ (vw ) and Cφ (w ) ̸ = Cφ (x) for every x ∈ NG (w ) \ {u, v}. Denote by A = {{l1 , l2 } : l1 ∈ L1 , l2 ∈ L2 , l1 ̸ = l2 }.
That is, A is a family of all possible sets {φ (uw ), φ (vw )} where φ (uw ) ̸ = φ (vw ), while φ (uw ) ∈ L1 and φ (vw ) ∈ L2 . Let m = |A|. We have
( m≥
) ⌈ 2k ⌉ + 3 2
,
and so 8m ≥ k2 + 10k + 24. Since 8n ≤ 8k − 16, we have 8(m − n) ≥ k2 + 2k + 40, and so m − n > 5. Hence there exist l1 ∈ L1 and l2 ∈ L2 such that (a) (b) (c) (d) (e)
l1 ̸ = l2 , σ (uw) ̸= l1 or σ (vw ) ̸= l2 , l1 ̸ ∈ (Cφ (u) ∪ Cσ (w )) − σ (uw ), l2 ̸ ∈ (Cφ (v ) ∪ Cσ (w )) − σ (vw ), Cσ (x) ̸ = (Cσ (w ) \ {σ (uw ), σ (vw )}) ∪ {l1 , l2 }, for every x ∈ NG (w ) \ {u, v}.
We now assign the color l1 to uw and l2 to vw . Since the sets Cσ (u) and Cσ (v ) were initially equivalent, after the colors of uw and vw are changed these sets differ. Now, as shown before, there exists at least one color that can be used to color uv so that an nde-coloring with 2k colors is produced. This is a contradiction, which completes the proof of the theorem. □ We use the statement of the following lemma in the bases step of the induction in the proof of Theorem 2. Lemma 9. If G is a normal graph with 2 ≤ χ ′ (G) ≤ 5, then χa′ (G) ≤ 2χ ′ (G). Please cite this article in press as: B. Vučković, Edge-partitions of graphs and their neighbor-distinguishing index, Discrete Mathematics (2017), http://dx.doi.org/10.1016/j.disc.2017.07.005.
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Proof. If χ ′ (G) = 2, then every component of G is either an even cycle or a path of length 2 or more. Zhang et al. [7] showed that χa′ (Ck ) ≤ 4 for every k ̸ = 5, and χa′ (Pn ) ≤ 3 for every n ≥ 2. Thus the statement holds when χ ′ (G) = 2. We conclude from Theorem 5 that χa′ (G) ≤ 2∆(G) when 3 ≤ ∆(G) ≤ 5. Since χ ′ (G) ≥ ∆(G) for every graph G, we have χa′ (G) ≤ 2χ ′ (G) when 3 ≤ χ ′ (G) ≤ 5. □ We also need the next theorem in the proof of Theorem 2. Theorem 10. Let G and F be normal graphs such that G ⊆ F and χ ′ (G) = 2. Let H = F [E(F ) \ E(G)], with Hj , j ∈ {1, . . . , l}, as the components of H. If Hj = (xj1 , xj2 , xj3 ) is a path of length 2 for every j ∈ {1, . . . , l}, with degG (xj1 ) = 2 and degG (xj2 ) = degG (xj3 ) = 0, then χa′ (F ) ≤ 4. Proof. We know that ∆(G) ≤ χ ′ (G), and hence ∆(G) = 2, while χa′ (G) ≤ 4, by Lemma 9. Let σ be an nde-coloring of G with colors from L = {1, 2, 3, 4}. We now define an nde-coloring φ of F with colors from L as an extension of σ . Let φ (e) = σ (e) for every e ∈ E(G). Next, let U be a set of vertices in F that have exactly three incident edges in F . Thus degG (u) = 2 and degH (u) = 1, for every u ∈ U. Let J be the subgraph of G induced by U, and let J1 , . . . , Jm be the components of J. Since ∆(G) = 2, Ji is either a path or a cycle, for every 1 ≤ i ≤ m. We now assign colors to the edges of H that are incident with vertices from I = Ji , for every 1 ≤ i ≤ m. We have two cases, depending on whether I is a path or a cycle. 1. I is a path (v1 , . . . , vn ), n ≥ 1. Denote by vi vi′ the edge from H incident with vi , for every 1 ≤ i ≤ n. By definition, degG (vi ) = 2 for every i ∈ {1, . . . , n}. Thus we assign to v1 v1′ any of the two colors not used by φ for the edges incident with v1 . Suppose now that vj−1 vj′−1 , where j ≤ n, is the last edge to which we assigned a color. Next, we assign to vj vj′ a color from L different from the colors of the two other edges incident with vj in G, so that Cφ (vj ) ̸ = Cφ (vj−1 ) holds. Since there are four colors available, while we avoid at most three, such a coloring is always possible. We continue with this procedure until j = n, and thus color the edges vi vi′ , for every 1 ≤ i ≤ n. 2. I is a cycle (v1 , . . . , vn , v1 ). Denote by vi vi′ the edge from H incident with vi , for every 1 ≤ i ≤ n. Since σ is an nde-coloring of G, we have that Cσ (v1 ) ̸ = Cσ (vn ). Let S = Cσ (v1 ) ∪ Cσ (vn ). Since σ (v1 vn ) ∈ Cσ (v1 ) ∩ Cσ (v2 ) and Cσ (v1 ) ̸ = Cσ (vn ), we have |S | = 3. Next, we assign to v1 v1′ the remaining color from L \ S. Suppose now that vj−1 vj′−1 , where j ≤ n, is the last edge to which we assigned a color. Next, we assign a color to vj vj′ . Again, to obtain Cφ (vj ) ̸ = Cφ (vj−1 ) we avoid at most three colors, while there are four colors available. We continue with this procedure until j = n, thus coloring all the edges vi vi′ , for 1 ≤ i ≤ n. When j = n, we have that Cφ (vn ) ̸ = Cφ (v1 ), since we chose the color for v1 v1′ in such a manner that Cφ (v1 ) contains two colors not in Cσ (vn ). In the procedure above edges xj1 xj2 from E(Hj ), with degG (xj1 ) = 2 and degG (xj2 ) = 0 were colored for every j ∈ {1, . . . , l}. It remains to color the edges xj2 xj3 from E(Hj ), with degG (xj2 ) = degG (xj3 ) = 0, for every j ∈ {1, . . . , l}. Assign to xj2 xj3 any color from L different from the color assigned to xj1 xj2 . Since xj2 xj3 is adjacent in F only to xj1 xj2 , while degF (xj1 ) = 3, degF (xj2 ) = 2 and degF (xj3 ) = 1, we have Cφ (xj1 ) ̸ = Cφ (xj2 ) and Cφ (xj2 ) ̸ = Cφ (xj3 ). After these edges are colored for every Hj , 1 ≤ j ≤ l, the coloring φ is an nde-coloring of F with colors from L, completing the proof. □ The following Petersen’s lemma [3] is used in the proof of Theorem 4. Lemma 11. If G is a 2k-regular graph for some k ≥ 1, then G is 2-factorable. 3. Proofs of Theorems 2 and 4 Proof of Theorem 2. We proceed by induction on χ ′ (G) = k. Since G is a normal graph, it follows that k ≥ 2. If ∆(G) ≤ 5 or k ≤ 5, the statement is true by Theorem 5 and Lemma 9. Thus the basis step of the induction is true. Suppose that the theorem is false. Among all graphs with χ ′ (G) = k and χa′ (G) > 2k, let G be a graph with the minimum number of edges. Clearly, G is connected, and ∆(G) ≥ 6 and k ≥ 6. We may assume that χa′ (H) ≤ 2χ ′ (H) for every graph H with χ ′ (H) = k − 2. If |E(G)| ≤ 2χ ′ (G), then trivially χa′ (G) ≤ 2χ ′ (G); thus |E(G)| > 2χ ′ (G). Let φ be a proper edge coloring of G using the colors from K = {1, . . . , k}, and let K1 = {1, 2} and K2 = {3, . . . , k}. We may assume that any edge colored j, 1 < j ≤ k, has at least one adjacent edge colored i, for every 1 ≤ i < j. Otherwise, while there exists an edge e ∈ E(G) colored j that has no adjacent edge colored i, for some 1 ≤ i < j, change the color of e to i. Suppose that there exists uv ∈ E(G) such that wG (uv ) ≤ ∆(G) + 2. Then, by Theorem 8, there exists a normal graph H with ∆(H) ≤ ∆(G), |E(H)| < |E(G)| and χa′ (H) > 2k, contradicting the assumption that G is a minimal counterexample. Therefore, we may assume that wG (uv ) > ∆(G) + 2 ≥ 8 for every edge uv ∈ E(G). Hence at least one of these two vertices, say u, is incident with at least five edges. Since u has at most two incident edges colored from K1 , it has at least three incident edges colored from K2 . An edge colored 1 that has no adjacent edges colored 2 we call a pendent edge. While there exists a pendent edge uv ∈ E(G), do the following:
• If uv has no adjacent edges colored k, change the color of uv to k. • If uv has an adjacent edge, say uw, colored k, we consider two cases: – If w has an incident edge colored 1, change the color of uw to k + 1, and the color of uv to k + 2. – If none of the edges incident with w is colored 1, change the color of uw to 1, and the color of uv to 2. Please cite this article in press as: B. Vučković, Edge-partitions of graphs and their neighbor-distinguishing index, Discrete Mathematics (2017), http://dx.doi.org/10.1016/j.disc.2017.07.005.
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Notice that, since every e ∈ E(G) initially colored k was adjacent to an edge colored i for every 1 ≤ i < k, edge e is adjacent to at least one edge colored 2. Thus e has at most one adjacent pendent edge. Therefore, upon the above procedure, any edge colored k + 1 has exactly one adjacent edge colored k + 2. Clearly, the described coloring is a proper edge coloring with k + 2 colors. In addition, the following is satisfied.
• Every edge colored from K1 has at least one adjacent edge colored from K1 . • Every edge colored from K2 has at least one adjacent edge colored from K2 . • Every edge colored k + 1 has exactly one adjacent edge colored k + 2, and every edge colored k + 2 has exactly one adjacent edge colored k + 1. • For every uw ∈ E(G) colored k + 1, one of its incident vertices, say w, is incident with edges colored 1 and 2, while u is not incident with any edge colored 1 or 2. Every edge colored k + 2 has no adjacent edge colored from K1 . Let E = E(G) and let:
• E1 denotes the subset of edges from E that are colored from the set K1 . • E2 denotes the subset of edges from E that are colored from the set K2 . • S denotes the subset of edges from E that are colored from the set {k + 1, k + 2}. Let G1 = G[E1 ], G2 = G[E2 ], H = G[S ] and F = G[E1 ∪ S ]. Note that G1 , G2 , and F are normal graphs, where χ ′ (G1 ) = 2 and χ (G2 ) ≤ k − 2. Thus by the induction hypothesis, χa′ (G2 ) ≤ 2k − 4. Graph H consists of components Hj , j ∈ {1, . . . , l}, where every Hj = (xj1 , xj2 , xj3 ) is a path of length 3, degG1 (xj1 ) = 2 and degG1 (xj2 ) = degG1 (xj3 ) = 0. According to Theorem 10, χa′ (F ) ≤ 4. Graphs F and G2 form an edge-partition of G. Hence by Lemma 6, χa′ (G) ≤ χa′ (F ) + χa′ (G2 ), so χa′ (G) ≤ 4 + 2k − 4 = 2χ ′ (G), which is a contradiction. □ ′
Proof of Theorem 4. We proceed by induction on k. According to Theorem 5(2), the statement is satisfied for k = 2. Thus the basis step of the induction holds. Assume that χa′ (H) ≤ 2s+1 for every graph H with ∆(H) ≤ 2s , 2 ≤ s < k. Let m = |E(G)|. The statement is obviously true for m ≤ 2k+1 . Thus we may assume that m > 2k+1 . Suppose that the statement is false. Then among all counterexamples, let G be a graph with the minimum number of edges. Thus ∆(G) = 2k , χa′ (G) > 2k+1 , and clearly G is connected. Suppose that there exists uv ∈ E(G) such that wG (uv ) ≤ 2k + 2. Then, by Theorem 8, there exists a normal graph H with ∆(H) ≤ 2k , |E(H)| < m and χa′ (H) > 2k+1 , contradicting the assumption that G is a minimal counterexample. Hence we may assume that wG (uv ) > 2k + 2 for every uv ∈ E(G), which implies that degG (u) ≥ 2k−1 + 2 or degG (v ) ≥ 2k−1 + 2. It is well-known that for any graph G, there exists a ∆(G)-regular graph F such that G ⊆ F . Let F be such a graph. According to Lemma 11, F is 2-factorable. Let {F1 , . . . , F2k−1 } be a 2-factorization of F , and let H1 =
⋃2k−1
⋃2k−2 i=1
Fi
k−1
and H2 = -regular graphs and E(F ) = E(H1 ) ∪ E(H2 ). Let G1 = H1 ∩ G i=2k−2 +1 Fi . Thus H1 and H2 are edge-disjoint 2 and G2 = H2 ∩ G. Since ∆(Hi ) = 2k−1 , for 1 ≤ i ≤ 2, we have ∆(Gi ) ≤ 2k−1 , for 1 ≤ i ≤ 2. Since for every edge from G at least one of its incident vertices has a degree at least 2k−1 + 2, a degree of such a vertex in Gi , for 1 ≤ i ≤ 2, is at least 2. Therefore, every edge in Gi has at least one adjacent edge in Gi , for 1 ≤ i ≤ 2. Thus both graphs G1 and G2 are normal, E(G1 ) ∩ E(G2 ) = ∅, E(G1 ) ∪ E(G2 ) = E(G), and ∆(Gi ) ≤ 2k−1 , for 1 ≤ i ≤ 2. Also, χa′ (H) ≤ 2k for every graph H with ∆(H) ≤ 2k−1 ; thus χa′ (G) ≤ 2k + 2k = 2k+1 , by Lemma 6. This produces a contradiction. □ Acknowledgments The author thanks the anonymous referees for their careful comments that improved the presentation of this paper. The author was supported by the grant III044006 of the Ministry of Education and Science of Serbia. References [1] [2] [3] [4] [5] [6]
S. Akbari, H. Bidkhori, N. Nosrati, r-strong edge colorings of graphs, Discrete Math. 306 (2006) 3005–3010. P.N. Balister, E. Gyori, J. Lehel, R.H. Schelp, Adjacent vertex distinguishing edge-colorings, SIAM J. Discrete Math. 21 (2007) 237–250. J. Petersen, Die Theorie der regularen Graphen, Acta Math. 15 (1891) 193–220. V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25–30 (in Russian). Y. Wang, W. Wang, J. Huo, Some bounds on the neighbor-distinguishing index of graphs, Discrete Math. 338 (2015) 2006–2013. L. Zhang, W. Wang, K.W. Lih, An improved upper bound on the adjacent vertex distinguishing chromatic index of a graph, Discrete Appl. Math. 162 (2014) 348–354. [7] Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623–626.
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Edge-partitions of graphs and their neighbor-distinguishing index Bojan Vučković Mathematical Institute, Serbian Academy of Science and Arts, Kneza Mihaila 36 (P.O. Box 367), 11001 Belgrade, Serbia
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Article history: Received 1 March 2017 Received in revised form 4 July 2017 Accepted 5 July 2017 Available online xxxx Keywords: Neighbor-distinguishing edge coloring Maximum degree Edge-partition
a b s t r a c t A proper edge coloring is neighbor-distinguishing if any two adjacent vertices have distinct sets consisting of colors of their incident edges. The minimum number of colors needed for a neighbor-distinguishing edge coloring is the neighbor-distinguishing index, denoted by χa′ (G). A graph is normal if it contains no isolated edges. Let G be a normal graph, and let ∆(G) and χ ′ (G) denote the maximum degree and the chromatic index of G, respectively. We modify the previously known techniques of edge-partitioning to prove that χa′ (G) ≤ 2χ ′ (G), which implies that χa′ (G) ≤ 2∆(G) + 2. This improves the result in Wang et al. (2015), which states that χa′ (G) ≤ 25 ∆(G) for any normal graph. We also prove that χa′ (G) ≤ 2∆(G) when ∆(G) = 2k , k is an integer with k ≥ 2. © 2017 Elsevier B.V. All rights reserved.
1. Introduction All graphs considered in this paper are simple and finite. Let V (G), E(G), and ∆(G) denote the vertex set, the edge set, and the maximum degree of a graph G, respectively. Let NG (v ) and degG (v ) = |NG (v )| denote the set of neighbors and the degree of a vertex ⋃mv in G, respectively. An edge-partition of a graph G into subgraphs G1 , . . . , Gm is a decomposition of G such that E(G) = i=1 E(Gi ) and E(Gi ) ∩ E(Gj ) = ∅ for any pair i ̸ = j. For a graph G and any S ⊆ E(G), the edge-induced subgraph G[S ] is the subgraph of G whose edge set is S and whose vertex set consists of all end vertices of the edges in S. A proper edge k-coloring of a graph G is a function φ : E(G) → {1, . . . , k} such that every two adjacent edges receive different colors. The chromatic index χ ′ (G) of a graph G is the minimum positive integer k for which G has a proper edge k-coloring. Given an edge k-coloring φ of G, we use Cφ (v ) to denote the set of colors assigned to the edges incident with v . The edge coloring φ is called neighbor-distinguishing (in some papers adjacent vertex distinguishing), or nde-coloring for short, if Cφ (u) ̸ = Cφ (v ) for any pair of adjacent vertices u and v . The neighbor-distinguishing index χa′ (G) of a graph G is the smallest integer k such that G has a k-nde-coloring. A graph G is normal if it contains no isolated edges. It is obvious that G has an nde-coloring if and only if G is normal; thus we consider only normal graphs when examining an nde-coloring. Zhang, Liu and Wang [7] introduced and investigated a neighbor-distinguishing edge coloring of graphs, where they proposed the following conjecture. Conjecture 1. If G is a connected normal graph different from a 5-cycle, then χa′ (G) ≤ ∆(G) + 2. Akbari, Bidkhori and Nosrati [1] proved that χa′ (G) ≤ 3∆(G) for any normal graph G. Zhang, Wang and Lih [6] proved a better upper bound, χa′ (G) ≤ 25 ∆(G) + 5. Wang, Wang and Huo [5] improved this result by showing that χa′ (G) ≤ 25 ∆(G). Our main result is the following improvement of the upper bound for χa′ . E-mail address: [email protected]. http://dx.doi.org/10.1016/j.disc.2017.07.005 0012-365X/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: B. Vučković, Edge-partitions of graphs and their neighbor-distinguishing index, Discrete Mathematics (2017), http://dx.doi.org/10.1016/j.disc.2017.07.005.
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Theorem 2. If G is a normal graph, then χa′ (G) ≤ 2χ ′ (G). According to the famous Vizing’s theorem [4], χ ′ (G) ≤ ∆(G)+1. Thus as an immediate consequence we have the following corollary. Corollary 3. If G is a normal graph, then χa′ (G) ≤ 2∆(G) + 2. Furthermore, χa′ (G) of a normal graph G with ∆(G) = 2k , k ∈ N, k > 2, does not exceed 2∆(G), as stated in the next theorem. Theorem 4. If G is a normal graph with ∆(G) ≤ 2k , where k is an integer, and k ≥ 2, then χa′ (G) ≤ 2k+1 . The proofs of Theorems 2 and 4 are deferred to Section 3. First, we prove the statements that we will use in the proofs of these two theorems. 2. Preliminaries Ballister et al. [2] proved the first part, while Wang et al. [5] proved the second and third parts of the following theorem. Theorem 5. Let G be a normal graph. 1. If ∆(G) ≤ 3, then χa′ (G) ≤ 5. 2. If ∆(G) ≤ 4, then χa′ (G) ≤ 8. 3. If ∆(G) ≤ 5, then χa′ (G) ≤ 10. The following lemma and theorem, proved in [6], are the main tools used in papers [6] and [5] to attain the upper bounds. Lemma 6. If normal graph G has an edge-partition into normal graphs G1 and G2 , then χa′ (G) ≤ χa′ (G1 ) + χa′ (G2 ). Theorem 7. Let G be a normal graph with ∆(G) ≥ 6. Then there is an edge-partition of G into normal graphs H1 and H2 , such that: 1. ∆(H1 ) ≤ 3, 2. ∆(H2 ) ≤ ∆(G) − 2. The bound χa′ (G) ≤ 52 ∆(G) was proved by repeatedly applying the theorem above, and relying on the bounds of χa′ (H) for graphs with ∆(H) ≤ 5. We propose a procedure that is also edge-partitioning of a graph into two normal graphs, and then make use of Lemma 6. The difference is that we show how to produce an edge-partition of a normal graph G that has χ ′ (G) = k into normal graphs H1 and H2 with χa′ (H1 ) ≤ 4 and χ ′ (H2 ) ≤ k − 2. Consequently, using the bounds from Theorem 5 we get that χa′ (G) ≤ 2χ ′ (G) for any normal graph G. Define a function wG : E(G) → N by wG (uv ) = deg(u) + deg(v ) for each uv ∈ E(G). We use the next theorem in the proofs of Theorems 2 and 4 to show that a minimal counterexample of a graph does not contain an edge uv with wG (uv ) ≤ ∆(G) + 2. Theorem 8. Assume that there exists a connected graph G with ∆(G) ≥ 4 and χa′ (G) > 2k where ∆(G) ≤ k. If wG (uv ) ≤ ∆(G) + 2 for some uv ∈ E(G), then there exists a normal graph H with ∆(H) ≤ ∆(G), |E(H)| < |E(G)|, and χa′ (H) > 2k. Proof. Suppose that the statement is false. Let uv be an edge with wG (uv ) ≤ ∆(G) + 2, and let H = G − uv . Clearly, ∆(H) ≤ ∆(G) and |E(H)| < |E(G)|. Thus if H is a normal graph, then χa′ (H) ≤ 2k. Denote by L the set of colors {1, . . . , 2k}. Assume first that H is not a normal graph. Since G is a normal graph, this means that one of the vertices u and v , say v , has only one adjacent vertex w in H, and degH (w ) = 1. Now let H ′ = G − vw . Then degH ′ (v ) = 1 and degH ′ (w ) = 0. Also, degH ′ (u) > 1 since G is a connected graph with ∆(G) ≥ 4. Thus H ′ is a normal graph. Then there exists an nde-coloring σ of H ′ with colors from L. Let φ be an edge coloring of G with φ (e) = σ (e) for every e ∈ E(H ′ ). Assign to vw any color from L different from φ (uv ), and if degG (u) = degG (v ) = 2, a color not in Cφ (u). Since |L| > 2, this can always be done; thus φ is an nde-coloring with not more than 2k colors. This produces a contradiction. Therefore, we may assume that H is a normal graph, implying that χa′ (H) ≤ 2k. Depending on whether vertices u and v have a common neighbor, we consider two cases. 1. NG (u) ∩ NG (v ) = ∅. Let H ′ be a graph obtained from G by contracting the edge uv , and denote by w the vertex obtained by identifying u and v . Since degH ′ (w ) = degG (u) + degG (v ) − 2 ≤ ∆(G), it follows that ∆(H ′ ) ≤ ∆(G). Hence |E(H ′ )| < |E(G)| and so χa′ (H ′ ) ≤ 2k. Let σ ′ be an nde-coloring of H ′ with colors from L. Next, define an edge coloring σ of H in the following manner: Please cite this article in press as: B. Vučković, Edge-partitions of graphs and their neighbor-distinguishing index, Discrete Mathematics (2017), http://dx.doi.org/10.1016/j.disc.2017.07.005.
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• for every x ∈ NH (u), assign σ (ux) = σ ′ (wx), • for every y ∈ NH (v ), assign σ (v y) = σ ′ (wy), • to any other edge from H, assign the color assigned by σ ′ to the corresponding edge from H ′ . Edge coloring σ is proper, but not necessarily an nde-coloring of H. Since all the edges incident with w in H ′ are assigned different colors by σ ′ , we have Cσ (u) ∩ Cσ (v ) = ∅. Next, we define an nde-coloring φ of G with colors from L, which produces a contradiction. To each edge of G except uv assign the color assigned by σ . Thus we are left to color the edge uv . Suppose that wG (uv ) < k + 2. Then there are at most k − 1 colors assigned to the edges adjacent to uv . Also, there are at most k − 1 colors to avoid for uv to obtain that Cφ (u) is different from Cφ (u′ ) for every u′ ∈ NG (u) − v , and Cφ (v ) is different from Cφ (v ′ ) for every v ′ ∈ NG (v ) − u. Hence we color uv with any of the remaining colors from L. Since Cσ (u) ̸ = Cσ (v ), φ is an nde-coloring of G with 2k colors, which is a contradiction. Hence we assume that wG (uv ) = k + 2. Let d = degH (u) = degG (u) − 1, L1 = {1, . . . , d}, and L2 = {1, . . . , k − d}. Denote by ui , where i ∈ L1 , the vertices from NG (u) − v , and by vj , where j ∈ L2 , the vertices from NG (v ) − u. We may assume, without loss of generality, that φ (uui ) = i for every i ∈ L1 , φ (vvj ) = d + j for every j ∈ L2 , and d ≤ 2k . If there exists a color, say l, from K = {k + 1, . . . , 2k} such that Cσ (u) ∪ {l} ̸ = Cφ (ui ) for every i ∈ L1 and Cσ (v ) ∪ {l} ̸ = Cφ (vj ) for every j ∈ L2 , color uv with l. The coloring φ produced is an nde-coloring, which is a contradiction. Therefore, we may assume that Cφ (ui ) = {1, . . . , d} ∪ {k + i} for every i ∈ L1 , and Cφ (vj ) = {d + 1, . . . , k} ∪ {k + d + j} for every j ∈ L2 . Next, we change the color of uu1 , keeping the edge coloring proper, and Cφ (u1 ) ̸ = Cφ (x) for every x ∈ NG (u1 ) − u. We know that Cσ (u) ⊂ Cφ (u1 ) and |Cφ (u1 )| = d + 1. In addition, |NG (u1 ) − u| = d and d ≤ 2k . Thus to keep the edge coloring proper, and Cφ (u1 ) ̸ = Cφ (x) for every x ∈ NG (u1 ) − u, we need to avoid for uu1 at most 2d + 1 ≤ k + 1 colors. We change the color of uu1 to any of the remaining k − 1 colors. At last, assign the color 1 to uv . Such a coloring φ is an nde-coloring of G with 2k colors, which is a contradiction. 2. NG (u) ∩ NG (v ) ̸ = ∅. Denote by w one of the common neighbors of u and v in H. Let σ be an nde-coloring of H with colors from L. Next, let φ be an edge coloring of G that assigns the same color as σ to every edge of G − uv . Since wG (uv ) ≤ k + 2 and NH (u) ∩ NH (v ) ̸ = ∅, we have |NH (u) ∪ NH (v )| < k. Hence there are at most k − 1 vertices in H that are adjacent to u or v , or both. In addition, there are at most k edges adjacent to uv in G. Thus we can choose at least one of the colors from L for uv , call it l, so that the coloring is proper, and also Cφ (u) ̸ = Cφ (u′ ) for every u′ ∈ NG (u) − v , and Cφ (v ) ̸ = Cφ (v ′ ) for every v ′ ∈ NG (v ) − u. If Cσ (u) ̸ = Cσ (v ), assign l to uv , and we obtain an nde-coloring φ . In the described coloring, at most 2k colors were used, which contradicts the assumption that χa′ (G) > 2k. Therefore, we may assume that Cσ (u) = Cσ (v ), which implies that degH (u) = degH (v ) = s and k ≥ 2s. We now change the colors of uw and vw in φ while keeping the coloring proper and paying attention that Cφ (w ) remains different from Cφ (x) for every x ∈ NG (w ) \ {u, v}. Let n = |NG (w ) \ {u, v}|. Since ∆(G) ≤ k, we have n ≤ k − 2. Let L1 = {σ (uw )} ∪ L \ (Cσ (u) ∪ Cσ (w )) and L2 = {σ (vw )} ∪ L \ (Cσ (v ) ∪ Cσ (w )). We have that k
|L1 | ≥ 2k − (k − 2 + s) + 1 = k − s + 3 ≥ ⌈ ⌉ + 3 2
and |L2 | ≥ ⌈ 2k ⌉ + 3. Next, we choose a color from L1 for uw , and a color from L2 for vw , so that φ (uw ) ̸ = φ (vw ) and Cφ (w ) ̸ = Cφ (x) for every x ∈ NG (w ) \ {u, v}. Denote by A = {{l1 , l2 } : l1 ∈ L1 , l2 ∈ L2 , l1 ̸ = l2 }.
That is, A is a family of all possible sets {φ (uw ), φ (vw )} where φ (uw ) ̸ = φ (vw ), while φ (uw ) ∈ L1 and φ (vw ) ∈ L2 . Let m = |A|. We have
( m≥
) ⌈ 2k ⌉ + 3 2
,
and so 8m ≥ k2 + 10k + 24. Since 8n ≤ 8k − 16, we have 8(m − n) ≥ k2 + 2k + 40, and so m − n > 5. Hence there exist l1 ∈ L1 and l2 ∈ L2 such that (a) (b) (c) (d) (e)
l1 ̸ = l2 , σ (uw) ̸= l1 or σ (vw ) ̸= l2 , l1 ̸ ∈ (Cφ (u) ∪ Cσ (w )) − σ (uw ), l2 ̸ ∈ (Cφ (v ) ∪ Cσ (w )) − σ (vw ), Cσ (x) ̸ = (Cσ (w ) \ {σ (uw ), σ (vw )}) ∪ {l1 , l2 }, for every x ∈ NG (w ) \ {u, v}.
We now assign the color l1 to uw and l2 to vw . Since the sets Cσ (u) and Cσ (v ) were initially equivalent, after the colors of uw and vw are changed these sets differ. Now, as shown before, there exists at least one color that can be used to color uv so that an nde-coloring with 2k colors is produced. This is a contradiction, which completes the proof of the theorem. □ We use the statement of the following lemma in the bases step of the induction in the proof of Theorem 2. Lemma 9. If G is a normal graph with 2 ≤ χ ′ (G) ≤ 5, then χa′ (G) ≤ 2χ ′ (G). Please cite this article in press as: B. Vučković, Edge-partitions of graphs and their neighbor-distinguishing index, Discrete Mathematics (2017), http://dx.doi.org/10.1016/j.disc.2017.07.005.
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Proof. If χ ′ (G) = 2, then every component of G is either an even cycle or a path of length 2 or more. Zhang et al. [7] showed that χa′ (Ck ) ≤ 4 for every k ̸ = 5, and χa′ (Pn ) ≤ 3 for every n ≥ 2. Thus the statement holds when χ ′ (G) = 2. We conclude from Theorem 5 that χa′ (G) ≤ 2∆(G) when 3 ≤ ∆(G) ≤ 5. Since χ ′ (G) ≥ ∆(G) for every graph G, we have χa′ (G) ≤ 2χ ′ (G) when 3 ≤ χ ′ (G) ≤ 5. □ We also need the next theorem in the proof of Theorem 2. Theorem 10. Let G and F be normal graphs such that G ⊆ F and χ ′ (G) = 2. Let H = F [E(F ) \ E(G)], with Hj , j ∈ {1, . . . , l}, as the components of H. If Hj = (xj1 , xj2 , xj3 ) is a path of length 2 for every j ∈ {1, . . . , l}, with degG (xj1 ) = 2 and degG (xj2 ) = degG (xj3 ) = 0, then χa′ (F ) ≤ 4. Proof. We know that ∆(G) ≤ χ ′ (G), and hence ∆(G) = 2, while χa′ (G) ≤ 4, by Lemma 9. Let σ be an nde-coloring of G with colors from L = {1, 2, 3, 4}. We now define an nde-coloring φ of F with colors from L as an extension of σ . Let φ (e) = σ (e) for every e ∈ E(G). Next, let U be a set of vertices in F that have exactly three incident edges in F . Thus degG (u) = 2 and degH (u) = 1, for every u ∈ U. Let J be the subgraph of G induced by U, and let J1 , . . . , Jm be the components of J. Since ∆(G) = 2, Ji is either a path or a cycle, for every 1 ≤ i ≤ m. We now assign colors to the edges of H that are incident with vertices from I = Ji , for every 1 ≤ i ≤ m. We have two cases, depending on whether I is a path or a cycle. 1. I is a path (v1 , . . . , vn ), n ≥ 1. Denote by vi vi′ the edge from H incident with vi , for every 1 ≤ i ≤ n. By definition, degG (vi ) = 2 for every i ∈ {1, . . . , n}. Thus we assign to v1 v1′ any of the two colors not used by φ for the edges incident with v1 . Suppose now that vj−1 vj′−1 , where j ≤ n, is the last edge to which we assigned a color. Next, we assign to vj vj′ a color from L different from the colors of the two other edges incident with vj in G, so that Cφ (vj ) ̸ = Cφ (vj−1 ) holds. Since there are four colors available, while we avoid at most three, such a coloring is always possible. We continue with this procedure until j = n, and thus color the edges vi vi′ , for every 1 ≤ i ≤ n. 2. I is a cycle (v1 , . . . , vn , v1 ). Denote by vi vi′ the edge from H incident with vi , for every 1 ≤ i ≤ n. Since σ is an nde-coloring of G, we have that Cσ (v1 ) ̸ = Cσ (vn ). Let S = Cσ (v1 ) ∪ Cσ (vn ). Since σ (v1 vn ) ∈ Cσ (v1 ) ∩ Cσ (v2 ) and Cσ (v1 ) ̸ = Cσ (vn ), we have |S | = 3. Next, we assign to v1 v1′ the remaining color from L \ S. Suppose now that vj−1 vj′−1 , where j ≤ n, is the last edge to which we assigned a color. Next, we assign a color to vj vj′ . Again, to obtain Cφ (vj ) ̸ = Cφ (vj−1 ) we avoid at most three colors, while there are four colors available. We continue with this procedure until j = n, thus coloring all the edges vi vi′ , for 1 ≤ i ≤ n. When j = n, we have that Cφ (vn ) ̸ = Cφ (v1 ), since we chose the color for v1 v1′ in such a manner that Cφ (v1 ) contains two colors not in Cσ (vn ). In the procedure above edges xj1 xj2 from E(Hj ), with degG (xj1 ) = 2 and degG (xj2 ) = 0 were colored for every j ∈ {1, . . . , l}. It remains to color the edges xj2 xj3 from E(Hj ), with degG (xj2 ) = degG (xj3 ) = 0, for every j ∈ {1, . . . , l}. Assign to xj2 xj3 any color from L different from the color assigned to xj1 xj2 . Since xj2 xj3 is adjacent in F only to xj1 xj2 , while degF (xj1 ) = 3, degF (xj2 ) = 2 and degF (xj3 ) = 1, we have Cφ (xj1 ) ̸ = Cφ (xj2 ) and Cφ (xj2 ) ̸ = Cφ (xj3 ). After these edges are colored for every Hj , 1 ≤ j ≤ l, the coloring φ is an nde-coloring of F with colors from L, completing the proof. □ The following Petersen’s lemma [3] is used in the proof of Theorem 4. Lemma 11. If G is a 2k-regular graph for some k ≥ 1, then G is 2-factorable. 3. Proofs of Theorems 2 and 4 Proof of Theorem 2. We proceed by induction on χ ′ (G) = k. Since G is a normal graph, it follows that k ≥ 2. If ∆(G) ≤ 5 or k ≤ 5, the statement is true by Theorem 5 and Lemma 9. Thus the basis step of the induction is true. Suppose that the theorem is false. Among all graphs with χ ′ (G) = k and χa′ (G) > 2k, let G be a graph with the minimum number of edges. Clearly, G is connected, and ∆(G) ≥ 6 and k ≥ 6. We may assume that χa′ (H) ≤ 2χ ′ (H) for every graph H with χ ′ (H) = k − 2. If |E(G)| ≤ 2χ ′ (G), then trivially χa′ (G) ≤ 2χ ′ (G); thus |E(G)| > 2χ ′ (G). Let φ be a proper edge coloring of G using the colors from K = {1, . . . , k}, and let K1 = {1, 2} and K2 = {3, . . . , k}. We may assume that any edge colored j, 1 < j ≤ k, has at least one adjacent edge colored i, for every 1 ≤ i < j. Otherwise, while there exists an edge e ∈ E(G) colored j that has no adjacent edge colored i, for some 1 ≤ i < j, change the color of e to i. Suppose that there exists uv ∈ E(G) such that wG (uv ) ≤ ∆(G) + 2. Then, by Theorem 8, there exists a normal graph H with ∆(H) ≤ ∆(G), |E(H)| < |E(G)| and χa′ (H) > 2k, contradicting the assumption that G is a minimal counterexample. Therefore, we may assume that wG (uv ) > ∆(G) + 2 ≥ 8 for every edge uv ∈ E(G). Hence at least one of these two vertices, say u, is incident with at least five edges. Since u has at most two incident edges colored from K1 , it has at least three incident edges colored from K2 . An edge colored 1 that has no adjacent edges colored 2 we call a pendent edge. While there exists a pendent edge uv ∈ E(G), do the following:
• If uv has no adjacent edges colored k, change the color of uv to k. • If uv has an adjacent edge, say uw, colored k, we consider two cases: – If w has an incident edge colored 1, change the color of uw to k + 1, and the color of uv to k + 2. – If none of the edges incident with w is colored 1, change the color of uw to 1, and the color of uv to 2. Please cite this article in press as: B. Vučković, Edge-partitions of graphs and their neighbor-distinguishing index, Discrete Mathematics (2017), http://dx.doi.org/10.1016/j.disc.2017.07.005.
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Notice that, since every e ∈ E(G) initially colored k was adjacent to an edge colored i for every 1 ≤ i < k, edge e is adjacent to at least one edge colored 2. Thus e has at most one adjacent pendent edge. Therefore, upon the above procedure, any edge colored k + 1 has exactly one adjacent edge colored k + 2. Clearly, the described coloring is a proper edge coloring with k + 2 colors. In addition, the following is satisfied.
• Every edge colored from K1 has at least one adjacent edge colored from K1 . • Every edge colored from K2 has at least one adjacent edge colored from K2 . • Every edge colored k + 1 has exactly one adjacent edge colored k + 2, and every edge colored k + 2 has exactly one adjacent edge colored k + 1. • For every uw ∈ E(G) colored k + 1, one of its incident vertices, say w, is incident with edges colored 1 and 2, while u is not incident with any edge colored 1 or 2. Every edge colored k + 2 has no adjacent edge colored from K1 . Let E = E(G) and let:
• E1 denotes the subset of edges from E that are colored from the set K1 . • E2 denotes the subset of edges from E that are colored from the set K2 . • S denotes the subset of edges from E that are colored from the set {k + 1, k + 2}. Let G1 = G[E1 ], G2 = G[E2 ], H = G[S ] and F = G[E1 ∪ S ]. Note that G1 , G2 , and F are normal graphs, where χ ′ (G1 ) = 2 and χ (G2 ) ≤ k − 2. Thus by the induction hypothesis, χa′ (G2 ) ≤ 2k − 4. Graph H consists of components Hj , j ∈ {1, . . . , l}, where every Hj = (xj1 , xj2 , xj3 ) is a path of length 3, degG1 (xj1 ) = 2 and degG1 (xj2 ) = degG1 (xj3 ) = 0. According to Theorem 10, χa′ (F ) ≤ 4. Graphs F and G2 form an edge-partition of G. Hence by Lemma 6, χa′ (G) ≤ χa′ (F ) + χa′ (G2 ), so χa′ (G) ≤ 4 + 2k − 4 = 2χ ′ (G), which is a contradiction. □ ′
Proof of Theorem 4. We proceed by induction on k. According to Theorem 5(2), the statement is satisfied for k = 2. Thus the basis step of the induction holds. Assume that χa′ (H) ≤ 2s+1 for every graph H with ∆(H) ≤ 2s , 2 ≤ s < k. Let m = |E(G)|. The statement is obviously true for m ≤ 2k+1 . Thus we may assume that m > 2k+1 . Suppose that the statement is false. Then among all counterexamples, let G be a graph with the minimum number of edges. Thus ∆(G) = 2k , χa′ (G) > 2k+1 , and clearly G is connected. Suppose that there exists uv ∈ E(G) such that wG (uv ) ≤ 2k + 2. Then, by Theorem 8, there exists a normal graph H with ∆(H) ≤ 2k , |E(H)| < m and χa′ (H) > 2k+1 , contradicting the assumption that G is a minimal counterexample. Hence we may assume that wG (uv ) > 2k + 2 for every uv ∈ E(G), which implies that degG (u) ≥ 2k−1 + 2 or degG (v ) ≥ 2k−1 + 2. It is well-known that for any graph G, there exists a ∆(G)-regular graph F such that G ⊆ F . Let F be such a graph. According to Lemma 11, F is 2-factorable. Let {F1 , . . . , F2k−1 } be a 2-factorization of F , and let H1 =
⋃2k−1
⋃2k−2 i=1
Fi
k−1
and H2 = -regular graphs and E(F ) = E(H1 ) ∪ E(H2 ). Let G1 = H1 ∩ G i=2k−2 +1 Fi . Thus H1 and H2 are edge-disjoint 2 and G2 = H2 ∩ G. Since ∆(Hi ) = 2k−1 , for 1 ≤ i ≤ 2, we have ∆(Gi ) ≤ 2k−1 , for 1 ≤ i ≤ 2. Since for every edge from G at least one of its incident vertices has a degree at least 2k−1 + 2, a degree of such a vertex in Gi , for 1 ≤ i ≤ 2, is at least 2. Therefore, every edge in Gi has at least one adjacent edge in Gi , for 1 ≤ i ≤ 2. Thus both graphs G1 and G2 are normal, E(G1 ) ∩ E(G2 ) = ∅, E(G1 ) ∪ E(G2 ) = E(G), and ∆(Gi ) ≤ 2k−1 , for 1 ≤ i ≤ 2. Also, χa′ (H) ≤ 2k for every graph H with ∆(H) ≤ 2k−1 ; thus χa′ (G) ≤ 2k + 2k = 2k+1 , by Lemma 6. This produces a contradiction. □ Acknowledgments The author thanks the anonymous referees for their careful comments that improved the presentation of this paper. The author was supported by the grant III044006 of the Ministry of Education and Science of Serbia. References [1] [2] [3] [4] [5] [6]
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Please cite this article in press as: B. Vučković, Edge-partitions of graphs and their neighbor-distinguishing index, Discrete Mathematics (2017), http://dx.doi.org/10.1016/j.disc.2017.07.005.