In this paper, we prove that a graph G with no Ks,s-subgraph and twin-width d has r-admissibility r-coloring numbers bounded from above by an exponential function of r can construct graphs achieving such dependency in r.

A vertex-coloring of a hypergraph is conflict-free, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let f(r,∆) be the smallest integer k such that each r-uniform hypergraph of maximum vertex degree ∆ has a conflict-free coloring with at most k colors. As shown by Tardos and Pach, similarly to a classical Brooks’ type theorem for hypergraphs, f(r,∆) ≤...

Motivated b y reliability considerations i n data deduplication for s torage s ystems, w e introduce the problem of flexible coloring. Given a hypergraph H and the number of allowable colors k, a exible coloring o f H is a n assignment o f o ne or more colors to each vertex s uch that, for each hyperedge, i t is possible to choose a color from each vertex's color list so that this hyperedge i s...

‎For a coloring $c$ of a graph $G$‎, ‎the edge-difference coloring sum and edge-sum coloring sum with respect to the coloring $c$ are respectively‎ ‎$sum_c D(G)=sum |c(a)-c(b)|$ and $sum_s S(G)=sum (c(a)+c(b))$‎, ‎where the summations are taken over all edges $abin E(G)$‎. ‎The edge-difference chromatic sum‎, ‎denoted by $sum D(G)$‎, ‎and the edge-sum chromatic sum‎, ‎denoted by $sum S(G)$‎, ‎a...

Generalizing previous models of hypergraph coloring — due to Voloshin, Drgas-Burchardt and Lazuka, and the present authors – in this paper we introduce and study the structure class that we call stably bounded hypergraphs. In this model, a hypergraph is viewed as a six-tuple H = (X, E , s, t,a, b), where s, t,a, b : E → N are given integer-valued functions on the edge set. A mapping φ : X → N i...

Let G = (V, E) be a simple, finite, undirected graph. Let k, r be positive integers. A set S  V (G) is called (k, r)-semi strongly stable set if | (u) S|  k, for all u V (G). A partition of V(G) into (k, r)-semi strongly stable sets is called (k, r)-semi strong coloring of G. The minimum order of a (k, r)-semi strong coloring of G is called (k, r)-semi strong chromatic number of G and it is...

s i n c e t h e l a t e 1960s, s e v e r a l c o n s t i t u t i v e m o d e l s h a v e b e e n d e v e l o p e d f o r s a n d s d e s c r i b i n g t h e i r s t r e s s-s t r a i n e h a v i o r. m o s t o f t h e s e m o d e l s w e r e e s t a b l i s h e d b a s e d o n a m e a s u r e o f t h e p l a s t i c w o r k a n d s o m e y i e l d c r i t e r i a, a m o n g t h e m b e i n g t ...

Let H s − → G denote that any s-coloring of E(H) contains a monochromatic G. The degree Ramsey number of a graph G, denoted by R∆(G, s), is min{∆(H) : H s − → G}. We consider degree Ramsey numbers where G is a fixed even cycle. Kinnersley, Milans, and West showed that R∆(C2k, s) ≥ 2s, and Kang and Perarnau showed that R∆(C4, s) = Θ(s 2). Our main result is that R∆(C6, s) = Θ(s 3/2) and R∆(C10, ...

We introduce a concept in graph coloring motivated by the popular Sudoku puzzle. Let G=(V,E) be of order n with chromatic number χ(G)=k and let S⊆V. C0 k-coloring induced subgraph G[S]. The is called an extendable if can extended to G. say that G uniquely smallest such G[S] which admits denoted sn(G). In this paper we initiate study parameter. first show parameter related list graphs. Section 2...