id: 06109932
dt: j
an: 06109932
au: Campêlo, Manoel; Corrêa, Ricardo C.; Moura, Phablo F.S.; Santos, Marcio
    C.
ti: On optimal $k$-fold colorings of webs and antiwebs.
so: Discrete Appl. Math. 161, No. 1-2, 60-70 (2013).
py: 2013
pu: Elsevier Science B.V. (North-Holland), Amsterdam
la: EN
cc: 
ut: ($k$-fold) graph coloring; (fractional) chromatic number; clique and stable
    set numbers; web and antiweb
ci: 
li: doi:10.1016/j.dam.2012.07.013
ab: Summary: A $k$-fold $x$-coloring of a graph is an assignment of (at least)
    $k$ distinct colors from the set $\{1,2,\dots ,x\}$ to each vertex such
    that any two adjacent vertices are assigned disjoint sets of colors.
    The smallest number $x$ such that $G$ admits a $k$-fold $x$-coloring is
    the $k$-th chromatic number of $G$, denoted by $χ_{k}(G)$. We
    determine the exact value of this parameter when $G$ is a web or an
    antiweb. Our results generalize the known corresponding results for odd
    cycles and imply necessary and sufficient conditions under which
    $χ_{k}(G)$ attains its lower and upper bounds based on clique and
    integer and fractional chromatic numbers. Additionally, we extend the
    concept of $χ$-critical graphs to $χ_{k}$-critical graphs. We
    identify the webs and antiwebs having this property, for every integer
    $k\ge 1$.
rv: