We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is s-chromatic if it is colorable in s colors and any coloring of it uses at least s colors. The forcing chromatic number Fχ(G) of an s-chromatic graph G is the smallest number of vertices which must be colored so that, with the restriction that s colors are used, every remaining vertex has its color...

The zero forcing number is a graph invariant introduced in order to study the minimum rank of the graph. In the first part of this paper, we first highlight that the computation of the zero forcing number of any directed graph (allowing loops) is NP-hard. Furthermore, we identify a class of directed trees for which the zero forcing number is computable in linear time. The second part of the pap...

We show that א2 ≤ b < g is consistent. This work is dedicated to James Baumgartner on the occasion of his

The zero forcing number is a graph invariant introduced in order to study the minimum rank of the graph. In the first part of this paper, we first highlight that the computation of the zero forcing number of any directed graph (allowing loops) is NP-hard. Furthermore, we identify a class of directed trees for which the zero forcing number is computable in linear time. The second part of the pap...

We prove that two basic questions on outer measure are undecid-able. First we show that consistently • every sup-measurable function f : R 2 −→ R is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functions f : R 2 −→ R the Cauchy problem y ′ = f (x, y), y(x 0) = y 0 has a unique almost-everywhere solution in the class AC l (R) o...

The anti-forcing number of a perfect matching M graph G is the minimum edges whose deletion results in subgraph with unique M, denoted by af(G,M). When plane bipartite graph, Lei et al. established minimax result: For any G, af(G,M) equals maximum M-alternating cycles where two either are disjoint or intersect only at M; hexagonal system, Fries number. In this paper we show that for every syste...

‎In this paper‎, ‎we investigate a damped Korteweg-de‎ ‎Vries equation with forcing on a periodic domain‎ ‎$mathbb{T}=mathbb{R}/(2pimathbb{Z})$‎. ‎We can obtain that if the‎ ‎forcing is periodic with small amplitude‎, ‎then the solution becomes‎ ‎eventually time-periodic.

For a connected graph G = (V,E) of order at least two, a total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dmt(G). A subset T of a minimum total detour monophonic set S of G is a forcing total...