44 research outputs found

    DP-3-coloring of planar graphs without certain cycles

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    DP-coloring is a generalization of list coloring, which was introduced by Dvo\v{r}\'{a}k and Postle [J. Combin. Theory Ser. B 129 (2018) 38--54]. Zhang [Inform. Process. Lett. 113 (9) (2013) 354--356] showed that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is 3-choosable. Liu et al. [Discrete Math. 342 (2019) 178--189] showed that every planar graph without 4-, 5-, 6- and 9-cycles is DP-3-colorable. In this paper, we show that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is DP-3-colorable, which generalizes these results. Yu et al. gave three Bordeaux-type results by showing that (i) every planar graph with the distance of triangles at least three and no 4-, 5-cycles is DP-3-colorable; (ii) every planar graph with the distance of triangles at least two and no 4-, 5-, 6-cycles is DP-3-colorable; (iii) every planar graph with the distance of triangles at least two and no 5-, 6-, 7-cycles is DP-3-colorable. We also give two Bordeaux-type results in the last section: (i) every plane graph with neither 5-, 6-, 8-cycles nor triangles at distance less than two is DP-3-colorable; (ii) every plane graph with neither 4-, 5-, 7-cycles nor triangles at distance less than two is DP-3-colorable.Comment: 16 pages, 4 figure

    List-coloring and sum-list-coloring problems on graphs

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    Graph coloring is a well-known and well-studied area of graph theory that has many applications. In this dissertation, we look at two generalizations of graph coloring known as list-coloring and sum-list-coloring. In both of these types of colorings, one seeks to first assign palettes of colors to vertices and then choose a color from the corresponding palette for each vertex so that a proper coloring is obtained. A celebrated result of Thomassen states that every planar graph can be properly colored from any arbitrarily assigned palettes of five colors. This result is known as 5-list-colorability of planar graphs. Albertson asked whether Thomassen\u27s theorem can be extended by precoloring some vertices which are at a large enough distance apart. Hutchinson asked whether Thomassen\u27s theorem can be extended by allowing certain vertices to have palettes of size less than five assigned to them. In this dissertation, we explore both of these questions and answer them in the affirmative for various classes of graphs. We also provide a catalog of small configurations with palettes of different prescribed sizes and determine whether or not they can always be colored from palettes of such sizes. These small configurations can be useful in reducing certain planar graphs to obtain more information about their structure. Additionally, we look at the newer notion of sum-list-coloring where the sum choice number is the parameter of interest. In sum-list-coloring, we seek to minimize the sum of varying sizes of palettes of colors assigned the vertices of a graph. We compute the sum choice number for all graphs on at most five vertices, present some general results about sum-list-coloring, and determine the sum choice number for certain graphs made up of cycles

    Coloring and constructing (hyper)graphs with restrictions

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    We consider questions regarding the existence of graphs and hypergraphs with certain coloring properties and other structural properties. In Chapter 2 we consider color-critical graphs that are nearly bipartite and have few edges. We prove a conjecture of Chen, Erdős, Gyárfás, and Schelp concerning the minimum number of edges in a “nearly bipartite” 4-critical graph. In Chapter 3 we consider coloring and list-coloring graphs and hypergraphs with few edges and no small cycles. We prove two main results. If a bipartite graph has maximum average degree at most 2(k−1), then it is colorable from lists of size k; we prove that this is sharp, even with an additional girth requirement. Using the same approach, we also provide a simple construction of graphs with arbitrarily large girth and chromatic number (first proved to exist by Erdős). In Chapter 4 we consider list-coloring the family of kth power graphs. Kostochka and Woodall conjectured that graph squares are chromatic-choosable, as a strengthening of the Total List Coloring Conjecture. Kim and Park disproved this stronger conjecture, and Zhu asked whether graph kth powers are chromatic-choosable for any k. We show that this is not true: we construct families of graphs based on affine planes whose choice number exceeds their chromatic number by a logarithmic factor. In Chapter 5 we consider the existence of uniform hypergraphs with prescribed degrees and codegrees. In Section 5.2, we show that a generalization of the graphic 2-switch is insufficient to connect realizations of a given degree sequence. In Section 5.3, we consider an operation on 3-graphs related to the octahedron that preserves codegrees; this leads to an inductive definition for 2-colorable triangulations of the sphere. In Section 5.4, we discuss the notion of fractional realizations of degree sequences, in particular noting the equivalence of the existence of a realization and the existence of a fractional realization in the graph and multihypergraph cases. In Chapter 6 we consider a question concerning poset dimension. Dorais asked for the maximum guaranteed size of a subposet with dimension at most d of an n-element poset. A lower bound of sqrt(dn) was observed by Goodwillie. We provide a sublinear upper bound

    Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies

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    We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi to B is constrained in a specified way.Comment: 26 pages, no figures. Updated presentatio

    List-colourings of near-outerplanar graphs

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    A list-colouring of a graph is an assignment of a colour to each vertex v from its own list L(v) of colours. Instead of colouring vertices we may want to colour other elements of a graph such as edges, faces, or any combination of vertices, edges and faces. In this thesis we will study several of these different types of list-colouring, each for the class of a near-outerplanar graphs. Since a graph is outerplanar if it is both K4-minor-free and K2,3-minor-free, then by a near-outerplanar graph we mean a graph that is either K4-minor-free or K2,3-minor-free. Chapter 1 gives an introduction to the area of graph colourings, and includes a review of several results and conjectures in this area. In particular, four important and interesting conjectures in graph theory are the List-Edge-Colouring Conjecture (LECC), the List-Total-Colouring Conjecture (LTCC), the Entire Colouring Conjecture (ECC), and the List-Square-Colouring Conjecture (LSCC), each of which will be discussed in Chapter 1. In Chapter 2 we include a proof of the LECC and LTCC for all near-outerplanar graphs. In Chapter 3 we will study the list-colouring of a near-outerplanar graph in which vertices and faces, edges and faces, or vertices, edges and face are to be coloured. The results for the case when all elements are to be coloured will prove the ECC for all near-outerplanar graphs. In Chapter 4 we will study the list-colouring of the square of a K4-minor-free graph, and in Chapter 5 we will study the list-colouring of the square of a K2,3-minor-free graph. In Chapter 5 we include a proof of the LSCC for all K2,3-minor-free graphs with maximum degree at least six

    Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

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    We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatoric

    Graph Colouring and Frequency Assignment

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    In this thesis we study some graph colouring problems which arise from mathematical models of frequency assignment in radiocommunications networks, in particular from models formulated by Hale and by Tesman in the 1980s. The main body of the thesis is divided into four chapters. Chapter 2 is the shortest, and is largely self-contained; it contains some early work on the frequency assignment problem, in which each edge of a graph is assigned a positive integer weight, and an assignment of integer colours to the vertices is sought in which the colours of adjacent vertices differ by at least the weight of the edge joining them. The remaining three chapters focus on problems which combine frequency assignment with list colouring, in which each vertex has a list of integers from which its colour must be chosen. In Chapter 3 we study list colourings where the colours of adjacent vertices must differ by at least a fixed integer s, and in Chapter 4 we add the additional restriction that the lists must be sets of consecutive integers. In both cases we investigate the required size of the lists so that a colouring can always be found. By considering the behaviour of these parameters as s tends to infinity, we formulate continuous analogues of the two problems, considering lists which are real intervals in Chapter 4, and arbitrary closed real sets in Chapter 5. This gives rise to two new graph invariants, the consecutive choosability ratio tau(G) and the choosability ratio sigma(G). We relate these to other known graph invariants, provide general bounds on their values, and determine specific values for various classes of graphs

    Local Perspectives on Planar Colouring

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    In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erdos, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this thesis, we introduce the concept of a local girth list assignment: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We state and prove a local list colouring theorem unifying the two theorems of Thomassen mentioned above. In particular, we show that if G is a planar graph and L is a list assignment for G such that |L(v)| ≥ 3 for all v in V(G); |L(v)| ≥ 4 for every vertex v contained in a 4-cycle; and |L(v)| ≥ 5 for every vertex v contained in a triangle, then G admits an L-colouring. Next, we generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvorak and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this thesis, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are critical for 3-correspondence colouring form hyperbolic families. Analogous results for list colouring were shown by Postle and Thomas. Using results on hyperbolic families proved by Postle and Thomas, we show further that this implies that locally planar graphs are 5-correspondence colourable; and, using results of Dvorak and Kawarabayashi, that there exist linear-time algorithms for the decidability of 5-correspondence colouring for embedded graphs. We show analogous results for 3-correspondence colouring graphs of girth at least five. Finally we show that, in general, slightly stronger hyperbolicity theorems imply that the associated family of planar graphs have exponentially many colourings. The existence of exponentially many colourings has been studied before for list-colouring: for instance, Thomassen showed (without using hyperbolicity) that planar graphs have exponentially many 5-list colourings, and that planar graphs of girth at least five have exponentially many 3-list colourings. Using our stronger hyperbolicity theorems, we prove that planar graphs of girth at least five have exponentially many 3-correspondence colourings, and that planar graphs have exponentially many 5-correspondence colourings. This latter result proves a conjecture of Langhede and Thomassen. As correspondence colouring generalizes list colouring, our theorems also provide new, independent proofs that there are exponentially many 5-list colourings of planar graphs, and 3-list colourings of planar graphs of girth at least five
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