Improper choosability of planar graphs has been widely studied. In particular, Škrekovski investigated the smallest integer gk such that every planar graph of girth at least gk is k-improper 2-choosable. He proved [9] that 6 ≤ g1 ≤ 9; 5 ≤ g2 ≤ 7; 5 ≤ g3 ≤ 6 and ∀k ≥ 4, gk = 5. In this paper, we study the greatest real M(k, l) such that every graph of maximum average degree less than M(k, l) is ...

Let F be a finite field with pc elements, let A be a n×n matrix over F , and let k be a positive integer. When is it true that for all X1, . . . , Xn ⊆ F with |Xi| = k+1 and for all Y1, . . . , Yn ⊆ F with |Yi| = k, there exist x ∈ X1×. . .×Xn and y ∈ (F \Y1)×. . .×(F \Yn) such that Ax = y? It is trivial that A has this property for k = pc − 1 if det(A) 6= 0. The permanent lemma of Noga Alon pr...

A graph is H-free if it has no induced subgraph isomorphic to H. We determine the computational complexity of the Choosability problem restricted to H-free graphs for every graph H that does not belong to {K1,3, P1 +P2, P1 +P3, P4}. We also show that if H is a linear forest, then the problem is fixed-parameter tractable when parameterized by k.

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). In this pair of papers, it is proved that every multicircuit C has total choosability (i.e., list total chromatic number) ch00(C ) equal to its ordinary total chromatic number 00(C ). In the present paper, the kernel method is used to prove this for every multicircuit that

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 $|l(v)|geq k$ ...

We introduce the concept of list total colorings and prove that every multigraph of maximum degree 3 is 5-total-choosable. We also show that the total-choosability of a graph of maximum degree 2 is equal to its total chromatic number.