A graph G is (1, 1)-colorable if its vertices can be partitioned into subsets V1 and V2 so that every vertex in G[Vi] has degree at most 1 for each i ∈ {1, 2}. We prove that every graph with maximum average degree at most 14 5 is (1, 1)-colorable. In particular, it follows that every planar graph with girth at least 7 is (1, 1)-colorable. On the other hand, we construct graphs with maximum aver...

Haj\'os conjectured that every graph containing no subdivision of the complete $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, maximum chromatic number such graphs $\Omega(s^2/\log s)$. We prove $O(s)$ colors are enough for a weakening this only requires monochromatic component to have bounded size (so-called clustered coloring). Our approach leads more res...

Traditional Adomian decomposition method (ADM) usually fails to solve singular initial value problems of Emden-Fowler type. To overcome this shortcoming, a new and effective modification of ADM that only requires calculation of the first Adomian polynomial is formally proposed in the present paper. Three singular initial value problems of Emden-Fowler type with , 1 α  , 2 and , 2  and have be...

A graph G is (d1, . . . , dl)-colorable if the vertex set of G can be partitioned into subsets V1, . . . , Vl such that the graph G[Vi] induced by the vertices of Vi has maximum degree at most di for all 1 6 i 6 l. In this paper, we focus on complexity aspects of such colorings when l = 2, 3. More precisely, we prove that, for any fixed integers k, j, g with (k, j) 6= (0, 0) and g > 3, either e...

There is a conjecture due to Shaoji 3], about uniquely vertex r-colorable graphs which states: \ If G is a uniquely vertex r-colorable graph with order n and size (r ? 1)n ? ? r 2 , then G contains a K r as its subgraph." In this paper for any natural number r we construct a K r-free, uniquely r-colorable graph with (r ? 1)n ? ? r 2 edges. These families of graphs are indeed counter examples to...

A mixed hypergraph consists of two families of edges: the C-edges and D-edges. In a coloring every C-edge has at least two vertices of the same color, while every D-edge has at least two vertices colored differently. The largest and smallest possible numbers of colors in a coloring are termed the upper and lower chromatic number, χ̄ and χ, respectively. A mixed hypergraph is called uniquely colo...

Starting from the work by Jones on representations of Thompson's group $F$, subgroups $F$ with interesting properties have been defined and studied. One these is called $3$-colorable subgroup $\mathcal{F}$, which consists elements whose ``regions'' given their tree diagrams are $3$-colorable. On other hand, in his representations, also gave a method to construct knots links $F$. Therefore it na...

Let G = (V,E) be a graph, let f : V (G)→ N, and let k ≥ 0 be an integer. A list-assignment L of G is a function that assigns to each vertex v of G a set (list) L(v) of colors: usually each color is a positive integer. We say that L is an f -assignment if |L(v)| = f(v) for all v ∈ V , and a k-assignment if |L(v)| = k for all v ∈ V . A coloring ofG is a function φ that assigns a color to each ver...

We discuss the computational complexity of determining the chromatic number of graphs without long induced paths. We prove NP-completeness of deciding whether a P 8-free graph is 5-colorable and of deciding whether a P 12-free graph is 4-colorable. Moreover, we give a polynomial time algorithm for deciding whether a P 5-free graph is 3-colorable.

In this paper uniquely list colorable graphs are studied. A graph G is called to be uniquely k–list colorable if it admits a k–list assignment from which G has a unique list coloring. The minimum k for which G is not uniquely k–list colorable is called the m–number of G. We show that every triangle–free uniquely colorable graph with chromatic number k+1, is uniquely k–list colorable. A bound fo...