In this paper, we study the Empire Problem, a generalization of the coloring problem to maps on two-dimensional compact surface whose genus is positive. Given a planar graph with a certain partition of the vertices into blocks of size r, for a given integer r, the problem consists of deciding if s colors are sufficient to color the vertices of the graph such that vertices of the same block have...

Let r, s ∈ N, r ≥ s, and P and Q be two additive and hereditary graph properties. A (P,Q)-total (r, s)-coloring of a graph G = (V,E) is a coloring of the vertices and edges of G by s-element subsets of Zr such that for each color i, 0 ≤ i ≤ r − 1, the vertices colored by subsets containing i induce a subgraph of G with property P, the edges colored by subsets containing i induce a subgraph of G...

treatment of bone metastases comprises over 10% of the workload of a radiation-oncology center. bone metastases produce severe pain and immobility, necessitate narcotic use, and reduce the quality of life. a good palliative treatment must be complete, free from side effects and fast; thus we decided to evaluate strontium-89 (89sr) effectivity for palliation of breast and prostate cancer bone me...

We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph: (1) The list chromatic number χl(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list Lv of colors, where each list has length at least r, there is a choice of one color from each vertex list Lv yielding a pr...

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

Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite numbers. For graphs $G$ and $H$, say $G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring contains monochromatic copy $H$. Let $H[t]$ denote the $t$-blowup Then number $G,H,r,$ $t$ is defined minimum $n$ such that $G[n] \overset{r}{\longrightarrow} H[t]$. proved upper lower bounds on are exponential in $t$...

It is a well-known proposition that every graph of chromatic number larger than t contains every tree with t edges. The ‘standard’ reasoning is that such a graph must contain a subgraph ofminimumdegree at least t . Bohman, Frieze, andMubayi noticed that, although this argument does not work for hypergraphs, it is still possible that the proposition holds for hypergraphs as well. Indeed, Loh rec...

For S ⊆ Z+ and k and r fixed positive integers, denote by f(S, k; r) the least positive integer n (if it exists) such that within every r-coloring of {1, 2, . . . , n} there must be a monochromatic sequence {x1, x2, . . . , xk} with xi − xi−1 ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S, k; r) for various choices of S, as well as upper and lower bounds on this function. In particular, we...