Wyniki wyszukiwania

1

Generalized Fractional Total Colorings of Complete Graph

100%

Karafová G.

Discussiones Mathematicae Graph Theory

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2013

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tom 33

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nr 4

665-676

EN

An additive and hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be two additive and hereditary graph properties and let r, s be integers such that r ≥ s Then an [...] fractional (P,Q)-total coloring of a finite graph G = (V,E) is a mapping f, which assigns an s-element subset of the set {1, 2, . . . , r} to each vertex and each edge, moreover, for any color i all vertices of color i induce a subgraph of property P, all edges of color i induce a subgraph of property Q and vertices and incident edges have assigned disjoint sets of colors. The minimum ratio [...] of an [...] - fractional (P,Q)-total coloring of G is called fractional (P,Q)-total chromatic number X″f,P,Q(G) = [...] Let k = sup{i : Ki+1 ∈ P} and l = sup{i Ki+1 ∈ Q}. We show for a complete graph Kn that if l ≥ k +2 then _X″f,P,Q(Kn) = [...] for a sufficiently large n.

2

A Note on Neighbor Expanded Sum Distinguishing Index

100%

Flandrin E., Li H., Marczyk A., Saclé J.-F., Woźniak M.

Discussiones Mathematicae Graph Theory

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2017

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tom 37

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nr 1

29-37

EN

A total k-coloring of a graph G is a coloring of vertices and edges of G using colors of the set [k] = {1, . . . , k}. These colors can be used to distinguish the vertices of G. There are many possibilities of such a distinction. In this paper, we consider the sum of colors on incident edges and adjacent vertices.

3

Generalized Fractional Total Colorings of Graphs

100%

Karafová G., Soták R.

Discussiones Mathematicae Graph Theory

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2015

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tom 35

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nr 3

463-473

EN

Let P and Q be additive and hereditary graph properties and let r, s be integers such that r ≥ s. Then an r/s -fractional (P,Q)-total coloring of a finite graph G = (V,E) is a mapping f, which assigns an s-element subset of the set {1, 2, . . . , r} to each vertex and each edge, moreover, for any color i all vertices of color i induce a subgraph with property P, all edges of color i induce a subgraph with property Q and vertices and incident edges have been assigned disjoint sets of colors. The minimum ratio of an r/s -fractional (P,Q)-total coloring of G is called fractional (P,Q)-total chromatic number χ″ƒ,P,Q(G) = r/ s . We show in this paper that χ″ƒ,P,Q of a graph G with o(V (G)) vertex orbits and o(E(G)) edge orbits can be found as a solution of a linear program with integer coefficients which consists only of o(V (G)) + o(E(G)) inequalities.

4

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Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles

100%

Cranston D.

Discussiones Mathematicae Graph Theory

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2009

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tom 29

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nr 1

163-178

EN

Let G be a planar graph with no two 3-cycles sharing an edge. We show that if Δ(G) ≥ 9, then χ'ₗ(G) = Δ(G) and χ''ₗ(G) = Δ(G)+1. We also show that if Δ(G) ≥ 6, then χ'ₗ(G) ≤ Δ(G)+1 and if Δ(G) ≥ 7, then χ''ₗ(G) ≤ Δ(G)+2. All of these results extend to graphs in the projective plane and when Δ(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with small degree-sum. For example, we prove that if G is a planar graph with no two 3-cycles sharing an edge and with Δ(G) ≥ 7, then G has an edge uv with d(u) ≤ 4 and d(u)+d(v) ≤ Δ(G)+2. All of our proofs yield linear-time algorithms that produce the desired colorings.

5

A note on total colorings of planar graphs without 4-cycles

100%

Wang P., Wu J.-L.

Discussiones Mathematicae Graph Theory

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2004

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tom 24

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nr 1

125-135

EN

Let G be a 2-connected planar graph with maximum degree Δ such that G has no cycle of length from 4 to k, where k ≥ 4. Then the total chromatic number of G is Δ +1 if (Δ,k) ∈ {(7,4),(6,5),(5,7),(4,14)}.

6

Fractional (P,Q)-Total List Colorings of Graphs

63%

Kemnitz A., Mihók P., Voigt M.

Discussiones Mathematicae Graph Theory

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2013

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tom 33

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nr 1

167-179

EN

Let r, s ∈ N, r ≥ s, and P and Q be two additive and hereditary graph properties. A (P,Q)-total (r, s)-coloring of a graph G = (V,E) is a coloring of the vertices and edges of G by s-element subsets of Zr such that for each color i, 0 ≤ i ≤ r − 1, the vertices colored by subsets containing i induce a subgraph of G with property P, the edges colored by subsets containing i induce a subgraph of G with property Q, and color sets of incident vertices and edges are disjoint. The fractional (P,Q)-total chromatic number χ′′ f,P,Q(G) of G is defined as the infimum of all ratios r/s such that G has a (P,Q)-total (r, s)-coloring. A (P,Q)-total independent set T = VT ∪ET ⊆ V ∪E is the union of a set VT of vertices and a set ET of edges of G such that for the graphs induced by the sets VT and ET it holds that G[VT ] ∈ P, G[ET ] ∈ Q, and G[VT ] and G[ET ] are disjoint. Let TP,Q be the set of all (P,Q)-total independent sets of G. Let L(x) be a set of admissible colors for every element x ∈ V ∪ E. The graph G is called (P,Q)-total (a, b)-list colorable if for each list assignment L with |L(x)| = a for all x ∈ V ∪E it is possible to choose a subset C(x) ⊆ L(x) with |C(x)| = b for all x ∈ V ∪ E such that the set Ti which is defined by Ti = {x ∈ V ∪ E : i ∈ C(x)} belongs to TP,Q for every color i. The (P,Q)- choice ratio chrP,Q(G) of G is defined as the infimum of all ratios a/b such that G is (P,Q)-total (a, b)-list colorable. We give a direct proof of χ′′ f,P,Q(G) = chrP,Q(G) for all simple graphs G and we present for some properties P and Q new bounds for the (P,Q)-total chromatic number and for the (P,Q)-choice ratio of a graph G.

Ograniczanie wyników

6 Discussiones Mathematicae Graph Theory

2 Karafová G.

1 Cranston D.

1 Flandrin E.

1 Kemnitz A.

1 Li H.

1 Marczyk A.

1 Mihók P.

1 Saclé J.-F.

1 Soták R.

1 Voigt M.

1 Wang P.

1 Woźniak M.

1 Wu J.-L.

1 2017

1 2015

2 2013

1 2009

1 2004

Generalized Fractional Total Colorings of Complete Graph

A Note on Neighbor Expanded Sum Distinguishing Index

Generalized Fractional Total Colorings of Graphs

Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles

A note on total colorings of planar graphs without 4-cycles

Fractional (P,Q)-Total List Colorings of Graphs