6 research outputs found

    Coloring and covering problems on graphs

    Get PDF
    The \emph{separation dimension} of a graph GG, written π(G)\pi(G), is the minimum number of linear orderings of V(G)V(G) such that every two nonincident edges are ``separated'' in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the \emph{fractional separation dimension} πf(G)\pi_f(G), which is the minimum of a/ba/b such that some aa linear orderings (repetition allowed) separate every two nonincident edges at least bb times. In contrast to separation dimension, we show fractional separation dimension is bounded: always πf(G)3\pi_f(G)\le 3, with equality if and only if GG contains K4K_4. There is no stronger bound even for bipartite graphs, since πf(Km,m)=πf(Km+1,m)=3mm+1\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}. We also compute πf(G)\pi_f(G) for cycles and some complete tripartite graphs. We show that πf(G)<2\pi_f(G)<\sqrt{2} when GG is a tree and present a sequence of trees on which the value tends to 4/34/3. We conjecture that when n=3mn=3m the K4K_4-free nn-vertex graph maximizing πf(G)\pi_f(G) is Km,m,mK_{m,m,m}. We also consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let π(G)\pi^\circ(G) be the number of circular orderings needed to separate all pairs, and let πf(G)\pi_f^\circ(G) be the fractional version. Among our results: (1) π(G)=1\pi^\circ(G)=1 if and only GG is outerplanar. (2) π(G)2\pi^\circ(G)\le2 when GG is bipartite. (3) π(Kn)log2log3(n1)\pi^\circ(K_n)\ge\log_2\log_3(n-1). (4) πf(G)32\pi_f^\circ(G)\le\frac{3}{2}, with equality if and only if K4GK_4\subseteq G. (5) πf(Km,m)=3m32m1\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}. A \emph{star kk-coloring} is a proper kk-coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to prove that every graph with maximum average degree less than 52\frac{5}{2} has an \Ifp, which is sharp and improves the result of Bu, Cranston, Montassier, Raspaud, and Wang (2009). As a corollary, we gain that every planar graph with girth at least 10 has a star 4-coloring. A proper vertex coloring of a graph GG is \emph{rr-dynamic} if for each vV(G)v\in V(G), at least min{r,d(v)}\min\{r,d(v)\} colors appear in NG(v)N_G(v). We investigate 33-dynamic versions of coloring and list coloring. We prove that planar and toroidal graphs are 3-dynamically 10-choosable, and this bound is sharp for toroidal graphs. Given a proper total kk-coloring cc of a graph GG, we define the \emph{sum value} of a vertex vv to be c(v)+uvE(G)c(uv)c(v) + \sum_{uv \in E(G)} c(uv). The smallest integer kk such that GG has a proper total kk-coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} χΣ(G)\chi''_{\Sigma}(G). Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that χΣ(G)Δ(G)+3\chi''_{\Sigma}(G)\leq \Delta(G)+3 for any simple graph with maximum degree Δ(G)\Delta(G). We prove this bound to be asymptotically correct by showing that χΣ(G)Δ(G)(1+o(1))\chi''_{\Sigma}(G)\leq \Delta(G)(1+o(1)). The main idea of our argument relies on Przyby{\l}o's proof (2014) for neighbor sum distinguishing edge-coloring

    8-star-choosability of a graph with maximum average degree less than 3

    No full text
    Graphs and AlgorithmsA proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) epsilon L(v). If G is L-star-colorable for any list assignment L with vertical bar L(v)vertical bar \textgreater= k for all v epsilon V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by X-s(l)(G), is the smallest integer k such that G is k-star-choosable. In this article, we prove that every graph G with maximum average degree less than 3 is 8-star-choosable. This extends a result that planar graphs of girth at least 6 are 8-star-choosable [A. Kundgen, C. Timmons, Star coloring planar graphs from small lists, J. Graph Theory, 63(4): 324-337, 2010]
    corecore