Colouring graphs with prescribed induced cycle lengths

Bert Randerath; Ingo Schiermeyer

Displaying similar documents to “Colouring graphs with prescribed induced cycle lengths”

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.

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...

4-cycle properties for characterizing rectagraphs and hypercubes

Khadra Bouanane, Abdelhafid Berrachedi (2017)

Czechoslovak Mathematical Journal

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A $(0, 2)$ -graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of $(0, λ)$ -graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free $(0, 2)$ -graph. $(0, 2)$ -graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in $(0, λ)$ -graphs and more specifically...

On generalized shift graphs

Christian Avart, Tomasz Łuczak, Vojtěch Rödl (2014)

Fundamenta Mathematicae

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In 1968 Erdős and Hajnal introduced shift graphs as graphs whose vertices are the k-element subsets of [n] = 1,...,n (or of an infinite cardinal κ ) and with two k-sets $A = a ₁, . . ., a_{k}$ and $B = b ₁, . . ., b_{k}$ joined if $a ₁ < a ₂ = b ₁ < a ₃ = b ₂ < \dots < a_{k} = b_{k - 1} < b_{k}$ . They determined the chromatic number of these graphs. In this paper we extend this definition and study the chromatic number of graphs defined similarly for other types of mutual position with respect to the underlying ordering. As a consequence of our result, we show the existence of a graph with...

Nearly complete graphs decomposable into large induced matchings and their applications

Noga Alon, Ankur Moitra, Benjamin Sudakov (2013)

Journal of the European Mathematical Society

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We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on $N$ vertices with $(\frac{N}{2}) - o (N^{2})$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1 - o (1)}$ . The second construction provides a covering of all edges of the complete graph $K_{N}$ by two graphs, each being the edge disjoint union of at most $N^{2 - δ}$ induced matchings, where $δ > 0, 076$ . This disproves (in a strong form) a conjecture of Meshulam,...

2-halvable complete 4-partite graphs

Dalibor Fronček (1998)

Discussiones Mathematicae Graph Theory

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A complete 4-partite graph $K_{m ₁, m ₂, m ₃, m ₄}$ is called d-halvable if it can be decomposed into two isomorphic factors of diameter d. In the class of graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with at most one odd part all d-halvable graphs are known. In the class of biregular graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with four odd parts (i.e., the graphs $K_{m, m, m, n}$ and $K_{m, m, n, n}$ ) all d-halvable graphs are known as well, except for the graphs $K_{m, m, n, n}$ when d = 2 and n ≠ m. We prove that such graphs are 2-halvable iff n,m ≥ 3. We also determine a new class of non-halvable graphs $K_{m ₁, m ₂, m ₃, m ₄}$ with three...

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 $c h (K_{4, 3 * 2, 2 * (k - 6), 1 * 3}) = k$ ...

Generalized list colourings of graphs

Mieczysław Borowiecki, Ewa Drgas-Burchardt, Peter Mihók (1995)

Discussiones Mathematicae Graph Theory

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We prove: (1) that $c h_{P} (G) - χ_{P} (G)$ can be arbitrarily large, where $c h_{P} (G)$ and $χ_{P} (G)$ are P-choice and P-chromatic numbers, respectively, (2) the (P,L)-colouring version of Brooks’ and Gallai’s theorems.

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...

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.

Independent cycles and paths in bipartite balanced graphs

Beata Orchel, A. Paweł Wojda (2008)

Discussiones Mathematicae Graph Theory

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Bipartite graphs G = (L,R;E) and H = (L’,R’;E’) are bi-placeabe if there is a bijection f:L∪R→ L’∪R’ such that f(L) = L’ and f(u)f(v) ∉ E’ for every edge uv ∈ E. We prove that if G and H are two bipartite balanced graphs of order |G| = |H| = 2p ≥ 4 such that the sizes of G and H satisfy ||G|| ≤ 2p-3 and ||H|| ≤ 2p-2, and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs. This result implies easily that...

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 $Q_{i}$ .

Paired domination in prisms of graphs

Christina M. Mynhardt, Mark Schurch (2011)

Discussiones Mathematicae Graph Theory

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The paired domination number $γ_{p r} (G)$ of a graph G is the smallest cardinality of a dominating set S of G such that ⟨S⟩ has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V(G)| independent edges. We provide characterizations of the following three classes of graphs: $γ_{p r} (π G) = 2 γ_{p r} (G)$ for all πG; $γ_{p r} (K ₂ ☐ G) = 2 γ_{p r} (G)$ ; $γ_{p r} (K ₂ ☐ G) = γ_{p r} (G)$ .

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...

On 𝓕-independence in graphs

Frank Göring, Jochen Harant, Dieter Rautenbach, Ingo Schiermeyer (2009)

Discussiones Mathematicae Graph Theory

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Let be a set of graphs and for a graph G let $α_{} (G)$ and $α *_{} (G)$ denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on $α_{} (G)$ and $α *_{} (G)$ are presented.

Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs

Éric Sopena (2012)

Discussiones Mathematicae Graph Theory

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The oriented chromatic number of an oriented graph $^{\to} G$ is the minimum order of an oriented graph $^{\to} H$ such that $^{\to} G$ admits a homomorphism to $^{\to} H$ . The oriented chromatic number of an undirected graph G is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph G, defined as the minimum order of an oriented graph $^{\to} U$ such that every orientation $^{\to} G$ of G admits a homomorphism to $^{\to} U$ . We give...

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...

A note on the independent domination number versus the domination number in bipartite graphs

Shaohui Wang, Bing Wei (2017)

Czechoslovak Mathematical Journal

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Let $γ (G)$ and $i (G)$ be the domination number and the independent domination number of $G$ , respectively. Rad and Volkmann posted a conjecture that $i (G) / γ (G) \leq Δ (G) / 2$ for any graph $G$ , where $Δ (G)$ is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than $Δ (G) / 2$ are provided as well.