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Displaying similar documents to “Persistency in the Traveling Salesman Problem on Halin graphs”

Intrinsic linking and knotting are arbitrarily complex

Erica Flapan, Blake Mellor, Ramin Naimi (2008)

Fundamenta Mathematicae

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We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, | l k ( Q i , Q j ) | α and | a ( Q i ) | α , where a ( Q i ) denotes the second coefficient of the Conway polynomial of Q i .

Edit distance measure for graphs

Tomasz Dzido, Krzysztof Krzywdziński (2015)

Czechoslovak Mathematical Journal

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In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for g ( n , l ) , the biggest number k guaranteeing that there exist l graphs on n vertices, each two having edit distance at least k . By edit distance of two graphs G , F we mean the number of edges needed to be added to or deleted from graph G to obtain graph F . This new extremal number g ( n , l ) is closely linked to the edit distance of graphs. Using probabilistic methods we show...

Note on improper coloring of 1 -planar graphs

Yanan Chu, Lei Sun, Jun Yue (2019)

Czechoslovak Mathematical Journal

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A graph G = ( V , E ) is called improperly ( d 1 , , d k ) -colorable if the vertex set V can be partitioned into subsets V 1 , , V k such that the graph G [ V i ] induced by the vertices of V i has maximum degree at most d i for all 1 i k . In this paper, we mainly study the improper coloring of 1 -planar graphs and show that 1 -planar graphs with girth at least 7 are ( 2 , 0 , 0 , 0 ) -colorable.

Edge-colouring of graphs and hereditary graph properties

Samantha Dorfling, Tomáš Vetrík (2016)

Czechoslovak Mathematical Journal

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Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph G , a hereditary graph property 𝒫 and l 1 we define χ 𝒫 , l ' ( G ) to be the minimum number of colours needed to properly colour the edges of G , such that any subgraph of G induced by edges coloured by (at most) l colours is in 𝒫 . We present a necessary and sufficient condition for the existence of χ 𝒫 , l ' ( G ) . We focus on edge-colourings of graphs with respect to the hereditary...

Saturation numbers for linear forests P 6 + t P 2

Jingru Yan (2023)

Czechoslovak Mathematical Journal

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A graph G is H -saturated if it contains no H as a subgraph, but does contain H after the addition of any edge in the complement of G . The saturation number, sat ( n , H ) , is the minimum number of edges of a graph in the set of all H -saturated graphs of order n . We determine the saturation number sat ( n , P 6 + t P 2 ) for n 10 3 t + 10 and characterize the extremal graphs for n > 10 3 t + 20 .

Remarks on D -integral complete multipartite graphs

Pavel Híc, Milan Pokorný (2016)

Czechoslovak Mathematical Journal

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A graph is called distance integral (or D -integral) if all eigenvalues of its distance matrix are integers. In their study of D -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on D -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs K p 1 , p 2 , p 3 with p 1 < p 2 < p 3 , and K p 1 , p 2 , p 3 , p 4 with p 1 < p 2 < p 3 < p 4 , as well as the infinite classes of distance integral...

Generalized connectivity of some total graphs

Yinkui Li, Yaping Mao, Zhao Wang, Zongtian Wei (2021)

Czechoslovak Mathematical Journal

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We study the generalized k -connectivity κ k ( G ) as introduced by Hager in 1985, as well as the more recently introduced generalized k -edge-connectivity λ k ( G ) . We determine the exact value of κ k ( G ) and λ k ( G ) for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case k = 3 .

Embedding products of graphs into Euclidean spaces

Mikhail Skopenkov (2003)

Fundamenta Mathematicae

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For any collection of graphs G , . . . , G N we find the minimal dimension d such that the product G × . . . × G N is embeddable into d (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and ( K 3 , 3 ) are not embeddable into 2 n , where K₅ and K 3 , 3 are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding L k O S 2 n - 1 , where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

Note on a conjecture for the sum of signless Laplacian eigenvalues

Xiaodan Chen, Guoliang Hao, Dequan Jin, Jingjian Li (2018)

Czechoslovak Mathematical Journal

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For a simple graph G on n vertices and an integer k with 1 k n , denote by 𝒮 k + ( G ) the sum of k largest signless Laplacian eigenvalues of G . It was conjectured that 𝒮 k + ( G ) e ( G ) + k + 1 2 , where e ( G ) is the number of edges of G . This conjecture has been proved to be true for all graphs when k { 1 , 2 , n - 1 , n } , and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all k ). In this note, this conjecture is proved to be true for all graphs when k = n - 2 , and for some new classes of graphs.

On distinguishing and distinguishing chromatic numbers of hypercubes

Werner Klöckl (2008)

Discussiones Mathematicae Graph Theory

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The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number χ D ( G ) of G. Extending these concepts to infinite graphs we prove that D ( Q ) = 2 and χ D ( Q ) = 3 , where Q denotes the hypercube of countable dimension. We also show that χ D ( Q ) = 4 , thereby completing the investigation of finite hypercubes with respect to χ D . Our...