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1
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Saturation Spectrum of Paths and Stars

100%
Faudree J., Faudree R. J., Gould R. J., Jacobson M. S., Thomas B. J.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
811-822
EN
A graph G is H-saturated if H is not a subgraph of G but the addition of any edge from G̅ to G results in a copy of H. The minimum size of an H-saturated graph on n vertices is denoted sat(n,H), while the maximum size is the well studied extremal number, ex(n,H). The saturation spectrum for a graph H is the set of sizes of H saturated graphs between sat(n,H) and ex(n,H). In this paper we completely determine the saturation spectrum of stars and we show the saturation spectrum of paths is continuous from sat(n, Pk) to within a constant of ex(n, Pk) when n is sufficiently large.
2
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The Saturation Number for the Length of Degree Monotone Paths

100%
Caro Y., Lauri J., Zarb C.
Discussiones Mathematicae Graph Theory
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2015
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tom 35
|
nr 3
557-569
EN
A degree monotone path in a graph G is a path P such that the sequence of degrees of the vertices in the order in which they appear on P is monotonic. The length (number of vertices) of the longest degree monotone path in G is denoted by mp(G). This parameter, inspired by the well-known Erdős- Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter mp(G). We call G saturated if, for every edge e added to G, mp(G + e) > mp(G), and we define h(n, k) to be the least possible number of edges in a saturated graph G on n vertices with mp(G) < k, while mp(G+e) ≥ k for every new edge e. We obtain linear lower and upper bounds for h(n, k), we determine exactly the values of h(n, k) for k = 3 and 4, and we present constructions of saturated graphs.
3
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Improved upper bounds for nearly antipodal chromatic number of paths

100%
Shen Y.-F., Zheng G.-P., He W.-J.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
159-174
EN
For paths Pₙ, G. Chartrand, L. Nebeský and P. Zhang showed that $ac'(Pₙ) ≤ \binom{n-2}{2} + 2$ for every positive integer n, where ac'(Pₙ) denotes the nearly antipodal chromatic number of Pₙ. In this paper we show that $ac'(Pₙ) ≤ \binom{n-2}{2} - n/2 - ⎣10/n⎦ + 7$ if n is even positive integer and n ≥ 10, and $ac'(Pₙ) ≤ \binom{n-2}{2} - (n-1)/2 - ⎣13/n⎦ + 8$ if n is odd positive integer and n ≥ 13. For all even positive integers n ≥ 10 and all odd positive integers n ≥ 13, these results improve the upper bounds for nearly antipodal chromatic number of Pₙ.
4
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A Note on Path Domination

88%
Alcón L.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 4
1021-1034
EN
We study domination between different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks). We succeeded in characterizing those graphs in which every uv-walk of one particular kind dominates every uv-walk of other specific kind. We thereby obtained new characterizations of standard graph classes like chordal, interval and superfragile graphs.
5
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Paths of low weight in planar graphs

88%
Fabrici I., Harant J., Jendrol' S.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 1
121-135
EN
The existence of paths of low degree sum of their vertices in planar graphs is investigated. The main results of the paper are: 1. Every 3-connected simple planar graph G that contains a k-path, a path on k vertices, also contains a k-path P such that for its weight (the sum of degrees of its vertices) in G it holds $w_G(P): = ∑_{u∈ V(P)} deg_G(u) ≤ (3/2)k² + 𝓞(k)$ 2. Every plane triangulation T that contains a k-path also contains a k-path P such that for its weight in T it holds $w_T(P): = ∑_{u∈ V(P)} deg_T(u) ≤ k² +13 k$ 3. Let G be a 3-connected simple planar graph of circumference c(G). If c(G) ≥ σ| V(G)| for some constant σ > 0 then for any k, 1 ≤ k ≤ c(G), G contains a k-path P such that $w_G(P) = ∑_{u∈ V(P)} deg_G(u) ≤ (3/σ + 3)k$.
6
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Generalized total colorings of graphs

88%
Borowiecki M., Kemnitz A., Marangio M., Mihók P.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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nr 2
209-222
EN
An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P,Q)-total coloring of a simple graph G is a coloring of the vertices V(G) and edges E(G) of G such that for each color i the vertices colored by i induce a subgraph of property P, the edges colored by i induce a subgraph of property Q and incident vertices and edges obtain different colors. In this paper we present some general basic results on (P,Q)-total colorings. We determine the (P,Q)-total chromatic number of paths and cycles and, for specific properties, of complete graphs. Moreover, we prove a compactness theorem for (P,Q)-total colorings.
7
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The strong isometric dimension of finite reflexive graphs

75%
Fitzpatrick S., Nowakowski R.
Discussiones Mathematicae Graph Theory
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2000
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tom 20
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nr 1
23-38
EN
The strong isometric dimension of a reflexive graph is related to its injective hull: both deal with embedding reflexive graphs in the strong product of paths. We give several upper and lower bounds for the strong isometric dimension of general graphs; the exact strong isometric dimension for cycles and hypercubes; and the isometric dimension for trees is found to within a factor of two.
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