An edge of a graph is light when the sum of the degrees of its endvertices is at most 13. The well-known Kotzig Theorem states that every 3-connected planar graph contains a light edge. Later, Borodin [1] extended this result to the class of planar graphs of minimum degree at least 3. We deal with generalizations of these results for planar graphs of minimum degree 2. Borodin, Kostochka and Woo...

In this paper, we extend the Grötzsch Theorem by proving that the clique hypergraph H(G) of every planar graph is 3-colorable. We also extend this result to list colorings by proving that H(G) is 4-choosable for every planar or projective planar graph G. Finally, 4-choosability of H(G) is established for the class of locally planar graphs on arbitrary surfaces.

Let ch(G) denote the choice number of a graph G (also called “list chromatic number” or “choosability” of G). Noel, Reed, and Wu proved the conjecture of Ohba that ch(G) = χ(G) when |V (G)| ≤ 2χ(G) + 1. We extend this to a general upper bound: ch(G) ≤ max{χ(G), ⌈(|V (G)| + χ(G)− 1)/3⌉}. Our result is sharp for |V (G)| ≤ 3χ(G) using Ohba’s examples, and it improves the best-known upper bound for...

In this paper, we extend the Grr otzsch Theorem by proving that the clique hypergraph H(G) of every planar graph is 3-colorable. We also extend this result to list colorings by proving that H(G) is 4-choosable for every planar or projective planar graph G. Finally, 4-choosability of H(G) is established for the class of locally planar graphs on arbitrary surfaces.

For a given $\varepsilon > 0$, we say that graph $G$ is $\varepsilon$-flexibly $k$-choosable if the following holds: for any assignment $L$ of color lists size $k$ on $V(G)$, preferred from list requested at set $R$ vertices, then least |R|$ these requests are satisfied by some $L$-coloring. We consider question flexible choosability in several classes with certain degeneracy conditions. charac...

A linear coloring is a proper coloring such that each pair of color classes induces a union of disjoint paths. We study the linear list chromatic number, denoted lcl(G), of sparse graphs. The maximum average degree of a graph G, denoted mad(G), is the maximum of the average degrees of all subgraphs of G. It is clear that any graph Gwithmaximumdegree ∆(G) satisfies lcl(G) ≥ ⌈∆(G)/2⌉ + 1. In this...

Suppose the edges and the vertices of a simple graph G are assigned k-element lists of real weights. By choosing a representative of each list, we specify a vertex colouring, where for each vertex its colour is defined as the sum of the weights of its incident edges and the weight of the vertex itself. How long lists ensures a choice implying a proper vertex colouring for any graph? Is there an...