Burning number of caterpillars

Liu, Huiqing; Hu, Xuejiao; Hu, Xiaolan

doi:10.1016/j.dam.2020.03.062

Cited by 21 publications

(16 citation statements)

References 9 publications

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“…As proven in [5], spiders satisfy the conjecture. As proven in [9] and independently in [6], caterpillars satisfy the conjecture. In [7], it was proven that any graph with minimum degree δ ≥ 23 satisfy the conjecture.…”

Section: Introductionsupporting

confidence: 56%

An improved bound on the burning number of graphs

Bonato¹,

Kamali²

2021

Preprint

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Section: Introductionsupporting

confidence: 56%

An improved bound on the burning number of graphs

Bonato¹,

Kamali²

2021

Preprint

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“…[6] conjectured that all connected graphs are well-burnable, and they showed that P n and C n are wellburnable. As showed in [17] and independently in [13], the caterpillar is well-burnable. Bonato and Lidbetter [8] proved that the generalized star is also well-burnable.…”

Section: Introductionmentioning

confidence: 70%

“…i=1 a i = t 2 for some t}. J 5 = {(13, 11, 1), (11,11,3), (22,13,1), (19,13,4), (17,13,6), (15,13,8), (13,13,10), (17,15,4), (15,15,6) (3,3), (4, 2), (5, 5)} and D 4 = {(2, 1), (4, 1), (4,3), (4,4), (6,1), (6,4), (6,5), (6,6), (7,7), (8,4), (8,6), (10,4)}.…”

Section: Introductionmentioning

confidence: 99%

Burning numbers of t-unicyclic graphs

Zhang

Liu

2021

Preprint

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Given a graph G, the burning number of G is the smallest integer k for which there are vertices x 1 , x 2 , . . . , x k such that (x 1 , x 2 , . . . , x k ) is a burning sequence of G. It has been shown that the graph burning problem is NP-complete, even for trees with maximum degree three, or linear forests. A t-unicyclic graph is a unicycle graph with exactly one vertex of degree greater than 2. In this paper, we first present the bounds for the burning number of t-unicyclic graphs, and then use the burning numbers of linear forests with at most three components to determine the burning number of all t-unicyclic graphs for t ≤ 2.

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“…Alongside improving the general upper bound for b(G), a common theme amongst papers in the graph burning literature is proving that specific graph classes are well-burnable. For instance, in addition to the previously mentioned graph classes, classes of trees including spiders and caterpillars have been proven to be well-burnable [7], [9], [12]. The contribution of this paper to the literature is to complement these findings but in a more general framework.…”

Section: Introductionmentioning

confidence: 84%

“…The condition that d > n−1 3 − 1 implies by [8] that G has a spanning caterpillar, say T . By [12], T is well-burnable. However by Theorem 3, b(G) ≤ b(T ) and so G is well-burnable.…”

Section: Corollarymentioning

confidence: 97%

Burning Graph Classes

Omar¹,

Rohilla²

2021

Preprint

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The Burning Number Conjecture, that a graph on n vertices can be burned in at most ⌈ √ n ⌉ rounds, has been of central interest for the past several years. Much of the literature toward its resolution focuses on two directions: tightening a general upper bound for the burning number, and proving the conjecture for specific graph classes. In the latter, most of the developments work within a specific graph class and exploit the intricacies particular to it. In this article, we broaden this approach by developing systematic machinery that can be used as test beds for asserting that graph classes satisfy the conjecture. We show how to use these to resolve the conjecture for several classes of graphs including triangle-free graphs with degree lower bounds, graphs with certain linear lower bounds on r-neighborhood sizes, all trees whose nonleaf vertices have degree at least 4, trees whose non-leaf vertices have degree at least 3 (on at least 81 vertices), trees whose non-leaf vertices are less than 2 3 concentrated in degree 2, and trees with a low concentration of high degree non-leaf vertices (the last two results holding for sufficiently many non-leaf vertices).

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Burning number of caterpillars

Cited by 21 publications

References 9 publications

An improved bound on the burning number of graphs

An improved bound on the burning number of graphs

Burning numbers of t-unicyclic graphs

Burning Graph Classes

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