190 research outputs found

### Full Orientability of the Square of a Cycle

Let D be an acyclic orientation of a simple graph G. An arc of D is called
dependent if its reversal creates a directed cycle. Let d(D) denote the number
of dependent arcs in D. Define m and M to be the minimum and the maximum number
of d(D) over all acyclic orientations D of G. We call G fully orientable if G
has an acyclic orientation with exactly k dependent arcs for every k satisfying
m <= k <= M. In this paper, we prove that the square of a cycle C_n of length n
is fully orientable except n=6.Comment: 7 pages, accepted by Ars Combinatoria on May 26, 201

### Acyclic list edge coloring of outerplanar graphs

AbstractAn acyclic list edge coloring of a graph G is a proper list edge coloring such that no bichromatic cycles are produced. In this paper, we prove that an outerplanar graph G with maximum degree Ξβ₯5 has the acyclic list edge chromatic number equal to Ξ

### The strong chromatic index of 1-planar graphs

The chromatic index $\chi'(G)$ of a graph $G$ is the smallest $k$ for which
$G$ admits an edge $k$-coloring such that any two adjacent edges have distinct
colors. The strong chromatic index $\chi'_s(G)$ of $G$ is the smallest $k$ such
that $G$ has a proper edge $k$-coloring with the condition that any two edges
at distance at most 2 receive distinct colors. A graph is 1-planar if it can be
drawn in the plane so that each edge is crossed by at most one other edge.
In this paper, we show that every graph $G$ with maximum average degree
$\bar{d}(G)$ has $\chi'_{s}(G)\le (2\bar{d}(G)-1)\chi'(G)$. As a corollary, we
prove that every 1-planar graph $G$ with maximum degree $\Delta$ has
$\chi'_{\rm s}(G)\le 14\Delta$, which improves a result, due to Bensmail et
al., which says that $\chi'_{\rm s}(G)\le 24\Delta$ if $\Delta\ge 56$

### A note on the adjacent vertex distinguishing total chromatic number of graphs

AbstractAn adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have different sets of colors. The minimum number of colors needed for such a total coloring of G is denoted by Οaβ³(G). In this note, we show that Οaβ³(G)β€2Ξ for any graph G with maximum degree Ξβ₯3

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