Decomposing graphs into a constant number of locally irregular subgraphs

Bensmail, Julien; Merker, Martin; Thomassen, Carsten

doi:10.1016/j.ejc.2016.09.011

Cited by 37 publications

(34 citation statements)

References 11 publications

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“…Consequently, the improvement in Theorem 4, also gives the following bound for general graphs (again analogously to the proof of Theorem 4.6 in [3]).…”

mentioning

confidence: 75%

New bounds for locally irregular chromatic index of bipartite and subcubic graphs

2018

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A graph is locally irregular if the neighbors of every vertex v have degrees distinct from the degree of v. A locally irregular edge-coloring of a graph G is an (improper) edge-coloring such that the graph induced on the edges of any color class is locally irregular. It is conjectured that 3 colors suffice for a locally irregular edge-coloring. Recently, Bensmail et al.(Bensmail, Merker, Thomassen: Decomposing graphs into a constant number of locally irregular subgraphs, European J. Combin., 60:124-134, 2017) settled the first constant upper bound for the problem to 328 colors. In this paper, using a combination of existing results, we present an improvement of the bounds for bipartite graphs and general graphs, setting the best upper bounds to 7 and 220, respectively. In addition, we also prove that 4 colors suffice for locally irregular edge-coloring of any subcubic graph.

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“…Consequently, the improvement in Theorem 4, also gives the following bound for general graphs (again analogously to the proof of Theorem 4.6 in [3]).…”

mentioning

confidence: 75%

New bounds for locally irregular chromatic index of bipartite and subcubic graphs

2018

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“…Namely, they conjectured that in fact every graph G without isolated edges and isolated triangles can be decomposed into 2 subgraphs fulfilling the 1-2-3 Conjecture with only weights 1 and 2 (i.e., admitting neighbour sum-distinguishing 2-edge-weightings). This interesting problem is independent of the 1-2-3 Conjecture itself, and is also partly related to the research from [9] -it is in particular known that this conjecture holds for bipartite graphs and subcubic graphs. See [7] for details and further observations concerning this new concept, and many other related problems and results.…”

Section: Discussionmentioning

confidence: 96%

“…In general it was also proved by Bensmail, Merker and Thomassen [9] that every connected graph which is not exceptional can be decomposed into (at most) 328 locally irregular subgraphs, what was then pushed down to 220 such subgraphs by Lužar, Przyby lo and Soták [22]. See also [5,6,9,22] for a number of partial and related results.…”

Section: Theorem 3 ([28]mentioning

confidence: 93%

Decomposability of graphs into subgraphs fulfilling the 1–2–3 Conjecture

Bensmail

Przybyło

2019

Discrete Applied Mathematics

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The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general. We prove that every d-regular graph, d ≥ 2, can be decomposed into at most 2 subgraphs (without isolated edges) fulfilling the 1-2-3 Conjecture if d / ∈ {10, 11, 12, 13, 15, 17}, and into at most 3 such subgraphs in the remaining cases. Additionally, we prove that in general every graph without isolated edges can be decomposed into at most 24 subgraphs fulfilling the 1-2-3 Conjecture, improving the previously best upper bound of 40. Both results are partly based on applications of the Lovász Local Lemma.

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“…Regarding the above-mentioned conjecture, Bensmail et al in [16] proved that every bipartite graph G which is not an odd length path satisfies χ ′ irr (G) ≤ 10. Also, they proved that if G admits a partitioning into locally irregular subgraphs, then χ ′ irr (G) ≤ 328.…”

Section: Conjecture 2 [12] For Every Non-exception Graphmentioning

confidence: 99%

“…Without loss of generality, assume that the graph G 1 is (α 1 , α 2 )-graph and the graph G 2 is (β 1 , β 2 )-graph. Since ∆(G) = 6, by attention to the structure of the graph B, with respect to the symmetry, the following cases for (α 1 α 2 , β 1 β 2 ) can be considered: (16,12), (15,12), (24,12), (14,12), (13,13). The graph G contains a copy of the complete bipartite graph K 1,6 , so the case (24,12) is not possible, also, the graph G contains a copy of complete bipartite graph K 3,6 , so the cases (16,12), (15,12), (14,12) are not possible.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Algorithmic complexity of weakly semiregular partitioning and the representation number

Ahadi

Dehghan

Mollahajiaghaei

2017

Theoretical Computer Science

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A graph G is weakly semiregular if there are two numbers a, b, such that the degree of every vertex is a or b. The weakly semiregular number of a graph G, denoted by wr(G), is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph. We present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two. On the other hand, we show that determining whether wr(G) = 2 for a given bipartite graph G with at most three numbers in its degree set is NP-complete. Among other results, for every tree T , we show thatThe semiregular number of a graph G, denoted by sr(G), is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is a semiregular graph. We prove that the semiregular number of a tree T is ⌈ ∆(T ) 2 ⌉. On the other hand, we show that determining whether sr(G) = 2 for a given bipartite graph G with ∆(G) ≤ 6 is NP-complete. In the second part of the work, we consider the representation number. A graph G has a representation modulo r if there exists an injective map ℓ : V (G) → Zr such that vertices v and u are adjacent if and only if |ℓ(u) − ℓ(v)| is relatively prime to r. The representation number, denoted by rep(G), is the smallest r such that G has a representation modulo r. Narayan and Urick conjectured that the determination of rep(G) for an arbitrary graph G is a difficult problem [38]. In this work, we confirm this conjecture and show that if NP = P, then for any ǫ > 0, there is no polynomial time (1 − ǫ) n 2 -approximation algorithm for the computation of representation number of regular graphs with n vertices. (Arash Ahadi) alidehghan@sce.carleton.ca (Ali Dehghan), mmol-laha@uwo.ca (Mohsen Mollahajiaghaei) . Conjecture 3 [17, 18] For every graph G, we have χ ′ reg−irr (G) ≤ 2. Recently, motivated by Conjecture 2 and Conjecture 3, Ahadi et al. in [5] presented the following conjecture. With Conjecture 4 they weaken Conjecture 2 and strengthen Conjecture 3.

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Decomposing graphs into a constant number of locally irregular subgraphs

Abstract: A graph is locally irregular if no two adjacent vertices have the same degree. The irregular chromatic index χ ′ irr

Cited by 37 publications

References 11 publications

New bounds for locally irregular chromatic index of bipartite and subcubic graphs

New bounds for locally irregular chromatic index of bipartite and subcubic graphs

Decomposability of graphs into subgraphs fulfilling the 1–2–3 Conjecture

Algorithmic complexity of weakly semiregular partitioning and the representation number

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