Generalized list colourings of graphs

Mieczysław Borowiecki; Ewa Drgas-Burchardt; Peter Mihók

Displaying similar documents to “Generalized list colourings 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 $D (Q_{ℵ} ₀) = 2$ and $χ_{D} (Q_{ℵ} ₀) = 3$ , where $Q_{ℵ} ₀$ denotes the hypercube of countable dimension. We also show that $χ_{D} (Q ₄) = 4$ , thereby completing the investigation of finite hypercubes with respect to $χ_{D}$ . Our...

Classes of hypergraphs with sum number one

Hanns-Martin Teichert (2000)

Discussiones Mathematicae Graph Theory

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A hypergraph ℋ is a sum hypergraph iff there are a finite S ⊆ ℕ⁺ and d̲,d̅ ∈ ℕ⁺ with 1 < d̲ < d̅ such that ℋ is isomorphic to the hypergraph $ℋ_{d ̲, d ̅} (S) = (V,)$ where V = S and $= e \subseteq S : d ̲ < | e | < d ̅ \land \sum_{v \in e} v \in S$ . For an arbitrary hypergraph ℋ the sum number(ℋ ) is defined to be the minimum number of isolatedvertices $w ₁, . . ., w_{σ} \notin V$ such that $ℋ \cup w ₁, . . ., w_{σ}$ is a sum hypergraph. For graphs it is known that cycles Cₙ and wheels Wₙ have sum numbersgreater than one. Generalizing these graphs we prove for the hypergraphs ₙ and ₙ that under a certain condition...

Persistency in the Traveling Salesman Problem on Halin graphs

Vladimír Lacko (2000)

Discussiones Mathematicae Graph Theory

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For the Traveling Salesman Problem (TSP) on Halin graphs with three types of cost functions: sum, bottleneck and balanced and with arbitrary real edge costs we compute in polynomial time the persistency partition $E_{A l l}$ , $E_{S o m e}$ , $E_{N o n e}$ of the edge set E, where: $E_{A l l}$ = e ∈ E, e belongs to all optimum solutions, $E_{N o n e}$ = e ∈ E, e does not belong to any optimum solution and $E_{S o m e}$ = e ∈ E, e belongs to some but not to all optimum solutions.

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 $k = 3$ .

On characterization of uniquely 3-list colorable complete multipartite graphs

Yancai Zhao, Erfang Shan (2010)

Discussiones Mathematicae Graph Theory

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For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_{2, 2, r}$ r ∈ 4,5,6,7,8, $K_{2, 3, 4}$ , $K_{1 * 4, 4}$ , $K_{1 * 4, 5}$ , $K_{1 * 5, 4}$ . Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_{2, 2, r}$ r ∈ 4,5,6,7,8, the others have been proved not...

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...

Equivalent classes for K₃-gluings of wheels

Halina Bielak (1998)

Discussiones Mathematicae Graph Theory

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In this paper, the chromaticity of K₃-gluings of two wheels is studied. For each even integer n ≥ 6 and each odd integer 3 ≤ q ≤ [n/2] all K₃-gluings of wheels $W_{q + 2}$ and $W_{n - q + 2}$ create an χ-equivalent class.

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 $(d_{1}, \dots, d_{k})$ -colorable if the vertex set $V$ can be partitioned into subsets $V_{1}, \dots, V_{k}$ such that the graph $G [V_{i}]$ induced by the vertices of $V_{i}$ has maximum degree at most $d_{i}$ for all $1 \leq i \leq k$ . 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.

Embedding products of graphs into Euclidean spaces

Mikhail Skopenkov (2003)

Fundamenta Mathematicae

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For any collection of graphs $G ₁, . . ., G_{N}$ we find the minimal dimension d such that the product $G ₁ \times . . . \times G_{N}$ is embeddable into $ℝ^{d}$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $(K_{3, 3}) ⁿ$ are not embeddable into $ℝ^{2 n}$ , 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 $L k O \to S^{2 n - 1}$ , where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

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...

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 $1 \leq k \leq n$ , denote by $𝒮_{k}^{+} (G)$ the sum of $k$ largest signless Laplacian eigenvalues of $G$ . It was conjectured that $𝒮_{k}^{+} (G) \leq e (G) + (\binom{k + 1}{2})$ , where $e (G)$ is the number of edges of $G$ . This conjecture has been proved to be true for all graphs when $k \in {1, 2, n - 1, n}$ , 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.

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, $| l k (Q_{i}, Q_{j}) | \geq α$ and $| a ₂ (Q_{i}) | \geq α$ , where $a ₂ (Q_{i})$ denotes the second coefficient of the Conway polynomial of $Q_{i}$ .

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 $l \geq 1$ 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...

On choosability of complete multipartite graphs $K_{4, 3 * t, 2 * (k - 2 t - 2), 1 * (t + 1)}$

Guo-Ping Zheng, Yu-Fa Shen, Zuo-Li Chen, Jin-Feng Lv (2010)

Discussiones Mathematicae Graph Theory

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A graph G is said to be chromatic-choosable if ch(G) = χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba’s conjecture is true if and only if it is true for complete multipartite graphs. In this paper we show that Ohba’s conjecture is true for complete multipartite graphs $K_{4, 3 * t, 2 * (k - 2 t - 2), 1 * (t + 1)}$ for all integers t ≥ 1 and k ≥ 2t+2, that is, $c h (K_{4, 3 * t, 2 * (k - 2 t - 2), 1 * (t + 1)}) = k$ , which extends the results $c h (K_{4, 3, 2 * (k - 4), 1 * 2}) = k$ given by Shen et al. (Discrete Math. 308 (2008) 136-143), and $c h (K_{4, 3 * 2, 2 * (k - 6), 1 * 3}) = k$ ...