Let G = (V,E) be a graph. A k-coloring for G is a function f : V → [k] such that f(u) 6= f(v) for all (u, v) ∈ E. In other words, a k-coloring is an assignment of vertices to k colors such that no edge is monochromatic. We say that a graph G is k-colorable if there exists a k-coloring for G. The chromatic number of G is the least k such that G is k-colorable. Given a k-colorable graph G, findin...

A k-chromatic graph G is called uniquely k-colorable if every k-coloring of the vertex set V of G induces the same partition of V into k color classes. There is an innnite class C of uniquely 4-colorable planar graphs obtained from the K 4 by repeatedly inserting new vertices of degree 3 in triangular faces. In this paper we are concerned with the well-known conjecture (see 6]) that every uniqu...

Recall that a (hyper)graph is d-degenerate if every of its nonempty subgraphs has a vertex of degree at most d. Every d-degenerate (hyper)graph is (easily) (d + 1)colorable. A (hyper)graph is almost d-degenerate if it is not d-degenerate, but every its proper subgraph is d-degenerate. In particular, if G is almost (k − 1)-degenerate, then after deleting any edge it is k-colorable. For k ≥ 2, we...

This is the first half of a two-part paper devoted to on-line 3-colorable graphs. Here on-line 3-colorable triangle-free graphs are characterized by a finite list of forbidden induced subgraphs. The key role in our approach is played by the family of graphs which are both triangleand (2K2 + K1)-free. Characterization of this family is given by introducing a bipartite modular decomposition conce...

A labeled graph  is bipartite if its vertex set  can be partitioned into two disjoint subsets  and ,  = ∪, such that every edge of  is of the form ( ), where  ∈  and  ∈ . Let  be a positive integer and  = {1 2     }. A labeled graph  is colorable if there exists a function  →  with the property that adjacent vertices must be colored differently. Clearly  is bipartit...

Given a fixed integer n, we prove Ramsey-type theorems for the classes of all finite ordered n-colorable graphs, finite n-colorable graphs, finite ordered n-chromatic graphs, and finite n-chromatic graphs.

for a given hypergraph $h$ with chromatic number $chi(h)$ and with no edge containing only one vertex, it is shown that the minimum number $l$ for which there exists a partition (also a covering) ${e_1,e_2,ldots,e_l}$ for $e(h)$, such that the hypergraph induced by $e_i$ for each $1leq ileq l$ is $k$-colorable, is $lceil log_{k} chi(h) rceil$.

A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. It is proved in this paper, if a uniquely edge-3-colorable, cubic graph G is cyclically 4-edgeconnected, but not cyclically 5-edge-connected, then G must contain a snark as a minor. This is an approach to a conjecture that every triangle free uniquely edge-3-colorable cubic graph must have the Petersen graph as...

Given a list L(v) for each vertex v, we say that the graph G is L-colorable if there is a proper vertex coloring of G where each vertex v takes its color from L(v). The graph is uniquely k-list colorable if there is a list assignment L such that |L(v)| = k for every vertex v and the graph has exactly one L-coloring with these lists. Mahdian and Mahmoodian [MM99] gave a polynomial-time character...

for a given graph $g=(v,e)$, let $mathscr l(g)={l(v) : vin v}$ be a prescribed list assignment. $g$ is $mathscr l$-$l(2,1)$-colorable if there exists a vertex labeling $f$ of $g$ such that $f(v)in l(v)$ for all $v in v$; $|f(u)-f(v)|geq 2$ if $d_g(u,v) = 1$; and $|f(u)-f(v)|geq 1$ if $d_g(u,v)=2$. if $g$ is $mathscr l$-$l(2,1)$-colorable for every list assignment $mathscr l$ with $|l(v)|geq k$ ...