Total weight choosability in Hypergraphs

A total weighting of the vertices and edges of a hypergraph is called vertex-coloring if the total weights of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this note we show that such a weighting is possible if every vertex has two, and every edge has three weights to choose from, extending a recent result on graphs to hypergraphs

A note on the simultaneous edge coloring

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a $(\Delta(G)+1)$-edge coloring where $\Delta(G)$ denotes the maximum degreee of $G$. Recently, Cabello raised the following question: given two graphs $G_1,G_2$ of maximum degree $\Delta$ on the same set of vertices $V$, is it possible to edge-color their (edge) union with $\Delta+2$ colors in such a way the restriction of $G$ to respectively the edges of $G_1$ and the edges of $G_2$ are edge-colorings? More generally, given $\ell$ graphs, how many colors do we need to color their union in such a way the restriction of the coloring to each graph is proper? In this short note, we prove that we can always color the union of the graphs $G_1,\ldots,G_\ell$ of maximum degree $\Delta$ with $\Omega(\sqrt{\ell} \cdot \Delta)$ colors and that there exist graphs for which this bound is tight up to a constant multiplicative factor. Moreover, for two graphs, we prove that at most $\frac 32 \Delta +4$ colors are enough which is, as far as we know, the best known upper bound

A connection between circular colorings and periodic schedules

AbstractWe show that there is a curious connection between circular colorings of edge-weighted digraphs and periodic schedules of timed marked graphs. Circular coloring of an edge-weighted digraph was introduced by Mohar [B. Mohar, Circular colorings of edge-weighted graphs, J. Graph Theory 43 (2003) 107–116]. This kind of coloring is a very natural generalization of several well-known graph coloring problems including the usual circular coloring [X. Zhu, Circular chromatic number: A survey, Discrete Math. 229 (2001) 371–410] and the circular coloring of vertex-weighted graphs [W. Deuber, X. Zhu, Circular coloring of weighted graphs, J. Graph Theory 23 (1996) 365–376]. Timed marked graphs G→ [R.M. Karp, R.E. Miller, Properties of a model for parallel computations: Determinancy, termination, queuing, SIAM J. Appl. Math. 14 (1966) 1390–1411] are used, in computer science, to model the data movement in parallel computations, where a vertex represents a task, an arc uv with weight cuv represents a data channel with communication cost, and tokens on arc uv represent the input data of task vertex v. Dynamically, if vertex u operates at time t, then u removes one token from each of its in-arc; if uv is an out-arc of u, then at time t+cuv vertex u places one token on arc uv. Computer scientists are interested in designing, for each vertex u, a sequence of time instants {fu(1),fu(2),fu(3),…} such that vertex u starts its kth operation at time fu(k) and each in-arc of u contains at least one token at that time. The set of functions {fu:u∈V(G→)} is called a schedule of G→. Computer scientists are particularly interested in periodic schedules. Given a timed marked graph G→, they ask if there exist a period p>0 and real numbers xu such that G→ has a periodic schedule of the form fu(k)=xu+p(k−1) for each vertex u and any positive integer k. In this note we demonstrate an unexpected connection between circular colorings and periodic schedules. The aim of this note is to provide a possibility of translating problems and methods from one area of graph coloring to another area of computer science

A note on one-sided interval edge colorings of bipartite graphs

For a bipartite graph $G$ with parts $X$ and $Y$, an $X$-interval coloring is a proper edge coloring of $G$ by integers such that the colors on the edges incident to any vertex in $X$ form an interval. Denote by $\chi'_{int}(G,X)$ the minimum $k$ such that $G$ has an $X$-interval coloring with $k$ colors. The author and Toft conjectured [Discrete Mathematics 339 (2016), 2628--2639] that there is a polynomial $P(x)$ such that if $G$ has maximum degree at most $\Delta$, then $\chi'_{int}(G,X) \leq P(\Delta)$. In this short note, we prove this conjecture; in fact, we prove that a cubic polynomial suffices. We also deduce some improved upper bounds on $\chi'_{int}(G,X)$ for bipartite graphs with small maximum degree

On Some Edge Folkman Numbers Small and Large

Edge Folkman numbers Fe(G1, G2; k) can be viewed as a generalization of more commonly studied Ramsey numbers. Fe(G1, G2; k) is defined as the smallest order of any Kk -free graph F such that any red-blue coloring of the edges of F contains either a red G1 or a blue G2. In this note, first we discuss edge Folkman numbers involving graphs Js = Ks − e, including the results Fe(J3, Kn; n + 1) = 2n − 1, Fe(J3, Jn; n) = 2n − 1, and Fe(J3, Jn; n + 1) = 2n − 3. Our modification of computational methods used previously in the study of classical Folkman numbers is applied to obtain upper bounds on Fe(J4, J4; k) for all k \u3e 4