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

Lužar, Borut; Przybyło, Jakub; Soták, Roman

doi:10.1007/s10878-018-0313-7

Cited by 21 publications

(13 citation statements)

References 8 publications

(18 reference statements)

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“…As Lužar, Przybyło and Soták proved that bipartite graphs with even size can even be decomposed into at most six locally irregular subgraphs [8], the exact same proof yields that every decomposable graph has irregular chromatic index at most 220. More precisely, this improved bound is a consequence of the following: 4,8]). For every connected k-degenerate graph G with even size, we have…”

Section: Introductionmentioning

confidence: 77%

“…For some classes of graphs, there is still some gap between the bound in Conjecture 1.1 and the best known one on the irregular chromatic index. In [4], decomposable bipartite graphs were proved to have irregular chromatic index at most 10, which was improved down to 7 in [8]. In [1], decomposable graphs with maximum degree 3 were proved to have irregular chromatic index at most 5, which was improved down to 4 in [8].…”

Section: Introductionmentioning

confidence: 99%

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Decomposing Degenerate Graphs into Locally Irregular Subgraphs

Bensmail

Dross

Nisse

2020

Graphs and Combinatorics

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A (undirected) graph is locally irregular if no two of its adjacent vertices have the same degree. A decomposition of a graph G into k locally irregular subgraphs is a partition E 1 ,. .. , E k of E(G) into k parts each of which induces a locally irregular subgraph. Not all graphs decompose into locally irregular subgraphs; however, it was conjectured that, whenever a graph does, it should admit such a decomposition into at most three locally irregular subgraphs. This conjecture was verified for a few graph classes in recent years. This work is dedicated to the decomposability of degenerate graphs with low degeneracy. Our main result is that decomposable k-degenerate graphs decompose into at most 3k + 1 locally irregular subgraphs, which improves on previous results whenever k ≤ 9. We improve this result further for some specific classes of degenerate graphs, such as bipartite cacti, k-trees, and planar graphs. Although our results provide only little progress towards the leading conjecture above, the main contribution of this work is rather the decomposition schemes and methods we introduce to prove these results.

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Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Decomposing Degenerate Graphs into Locally Irregular Subgraphs

Bensmail

Dross

Nisse

2020

Graphs and Combinatorics

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“…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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“…Also, they proved that if G admits a partitioning into locally irregular subgraphs, then χ ′ irr (G) ≤ 328. Recently, Lužar et al improved the previous bound for bipartite graphs and general graphs to 7 and 220, respectively [36]. For more information about this conjecture see [41].…”

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

confidence: 98%

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

Cited by 21 publications

References 8 publications

Decomposing Degenerate Graphs into Locally Irregular Subgraphs

Decomposing Degenerate Graphs into Locally Irregular Subgraphs

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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