Path spectra for trees

Chen, Guantao; Faudree, Ralph J.; Šoltés, ubomír

doi:10.1016/j.disc.2010.08.010

Cited by 3 publications

(5 citation statements)

References 6 publications

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“…It is not difficult to see that all sets of even integers are path spectra. In fact, all such sets of positive integers can be realized by trees as shown in [1]. To the contrary, we will show that for each k ≥ 3 there are infinitely many sets of odd integers which are not path spectra.…”

Section: Introductionmentioning

confidence: 93%

“…Moreover, we assume r 1 is the minimum and s 1 is the maximum with respect to the above choices. Since B is 2-connected, r 1 < s 1 …”

Section: Claim 45 |N C (G − V (C))| = 1 (That Is G − V (C) Has Examentioning

confidence: 99%

“…Since k ≥ 4, we assume, without loss of generality, that there are two maximal paths Q 1 , Q 2 in B 1 such that 1 …”

Section: Case 1 G Is a Squashmentioning

confidence: 99%

“…It is readily seen that ps(G) = {2m 1 It is readily seen that ps(G) = {2m 1 , m 2 }. Thus, {a, b} is an absolute path spectrum if one of a and b is even.…”

Section: Introductionmentioning

confidence: 96%

“…A connected graph G is named handcuff if G contains two nontrivial blocks B 1 and B 2 and these two blocks are connected by a path P internally disjoint from both B 1 and B 2 . Note that P may be a single vertex, in this case, B 1 and B 2 share a common vertex. …”

Section: Introductionmentioning

confidence: 99%

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Non‐path spectrum sets

Chen

Faudree

et al. 2008

Journal of Graph Theory

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A path of a graph is called maximal if it is not a proper subpath of any other path of the graph. The path spectrum of a graph G, denoted by ps (G), is the set of lengths of all maximal paths in the graph. A set S of positive integers is called a path spectrum if there is a connected graph G such that ps(G) = S. Jacobson et al. showed that all sets of positive integers with cardinality of 1 or 2 are path spectrum sets. Their results raised the question of whether all sets of positive integers are path spectra. We show that, for every positive integer k ≥ 3, there are infinitely many sets of k positive integers which are not path spectra. A set S of positive integers is called an absolute path spectrum if there are infinitely many connected

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Section: Introductionmentioning

confidence: 93%

“…Moreover, we assume r 1 is the minimum and s 1 is the maximum with respect to the above choices. Since B is 2-connected, r 1 < s 1 …”

Section: Claim 45 |N C (G − V (C))| = 1 (That Is G − V (C) Has Examentioning

confidence: 99%

“…Since k ≥ 4, we assume, without loss of generality, that there are two maximal paths Q 1 , Q 2 in B 1 such that 1 …”

Section: Case 1 G Is a Squashmentioning

confidence: 99%

“…It is readily seen that ps(G) = {2m 1 It is readily seen that ps(G) = {2m 1 , m 2 }. Thus, {a, b} is an absolute path spectrum if one of a and b is even.…”

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations