184 research outputs found

    On the Complexity of Digraph Colourings and Vertex Arboricity

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    It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular pp-colouring is NP-complete for all rational p>1p>1. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most pp is NP-complete for every rational p>1p>1. Deciding if the fractional dichromatic number of a digraph is at most pp is NP-complete for every p>1,p≠2p>1, p \neq 2. Deciding if the circular vertex arboricity of a graph is at most pp is NP-complete for every rational p>1p>1. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur

    A note on circular chromatic number of graphs with large girth and similar problems

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    In this short note, we extend the result of Galluccio, Goddyn, and Hell, which states that graphs of large girth excluding a minor are nearly bipartite. We also prove a similar result for the oriented chromatic number, from which follows in particular that graphs of large girth excluding a minor have oriented chromatic number at most 55, and for the ppth chromatic number χp\chi_p, from which follows in particular that graphs GG of large girth excluding a minor have χp(G)≤p+2\chi_p(G)\leq p+2

    Distance-two labelings of digraphs

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    For positive integers j≥kj\ge k, an L(j,k)L(j,k)-labeling of a digraph DD is a function ff from V(D)V(D) into the set of nonnegative integers such that ∣f(x)−f(y)∣≥j|f(x)-f(y)|\ge j if xx is adjacent to yy in DD and ∣f(x)−f(y)∣≥k|f(x)-f(y)|\ge k if xx is of distant two to yy in DD. Elements of the image of ff are called labels. The L(j,k)L(j,k)-labeling problem is to determine the λ⃗j,k\vec{\lambda}_{j,k}-number λ⃗j,k(D)\vec{\lambda}_{j,k}(D) of a digraph DD, which is the minimum of the maximum label used in an L(j,k)L(j,k)-labeling of DD. This paper studies λ⃗j,k\vec{\lambda}_{j,k}- numbers of digraphs. In particular, we determine λ⃗j,k\vec{\lambda}_{j,k}- numbers of digraphs whose longest dipath is of length at most 2, and λ⃗j,k\vec{\lambda}_{j,k}-numbers of ditrees having dipaths of length 4. We also give bounds for λ⃗j,k\vec{\lambda}_{j,k}-numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining λ⃗j,1\vec{\lambda}_{j,1}-numbers of ditrees whose longest dipath is of length 3.Comment: 12 pages; presented in SIAM Coference on Discrete Mathematics, June 13-16, 2004, Loews Vanderbilt Plaza Hotel, Nashville, TN, US

    An extensive English language bibliography on graph theory and its applications, supplement 1

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    Graph theory and its applications - bibliography, supplement

    Digraph Coloring Games and Game-Perfectness

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    In this thesis the game chromatic number of a digraph is introduced as a game-theoretic variant of the dichromatic number. This notion generalizes the well-known game chromatic number of a graph. An extended model also takes into account relaxed colorings and asymmetric move sequences. Game-perfectness is defined as a game-theoretic variant of perfectness of a graph, and is generalized to digraphs. We examine upper and lower bounds for the game chromatic number of several classes of digraphs. In the last part of the thesis, we characterize game-perfect digraphs with small clique number, and prove general results concerning game-perfectness. Some results are verified with the help of a computer program that is discussed in the appendix

    An extensive English language bibliography on graph theory and its applications

    Get PDF
    Bibliography on graph theory and its application
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