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- W2963212801 abstract "The chromatic number of a graph $G$, denoted by $chi(G)$, is the minimum $k$such that $G$ admits a $k$-coloring of its vertex set in such a way that eachcolor class is an independent set (a set of pairwise non-adjacent vertices).The dichromatic number of a digraph $D$, denoted by $chi_A(D)$, is the minimum$k$ such that $D$ admits a $k$-coloring of its vertex set in such a way thateach color class is acyclic. In 1976, Bondy proved that the chromatic number of a digraph $D$ is at mostits circumference, the length of a longest cycle. Given a digraph $D$, we will construct three different graphs whose chromaticnumbers bound $chi_A(D)$. Moreover, we prove: i) for integers $kgeq 2$, $sgeq 1$ and $r_1, ldots,r_s$ with $kgeq r_igeq 0$ and $r_ineq 1$ for each $iin[s]$, that if allcycles in $D$ have length $r$ modulo $k$ for some $rin{r_1,ldots,r_s}$,then $chi_A(D)leq 2s+1$; ii) if $D$ has girth $g$ and there are integers $k$and $p$, with $kgeq g-1geq pgeq 1$ such that $D$ contains no cycle of length$r$ modulo $lceil frac{k}{p} rceil p$ for each $rin{-p+2,ldots,0,ldots,p}$, then $chi_A (D)leq lceil frac{k}{p} rceil$;iii) if $D$ has girth $g$, the length of a shortest cycle, and circumference$c$, then $chi_A(D)leq lceil frac{c-1}{g-1} rceil +1$, which improves,substantially, the bound proposed by Bondy. Our results show that if we havemore information about the lengths of cycles in a digraph, then we can improvethe bounds for the dichromatic number known until now." @default.
- W2963212801 created "2019-07-30" @default.
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- W2963212801 date "2018-10-01" @default.
- W2963212801 modified "2023-10-18" @default.
- W2963212801 title "New Bounds for the Dichromatic Number of a Digraph" @default.
- W2963212801 hasPublicationYear "2018" @default.
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