Variations on the Roy-Gallai theorem

Werra, Dominique de; Hansen, Pierre

doi:10.1007/s10288-004-0043-9

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Long Rainbow Paths In χ(G)-colorings1

A(v) < A(u) < R(u) < R(v)1

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2005

2011

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Cited by 5 publications

(2 citation statements)

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“…Proof. Let c be a χ-coloring of G. As a consequence of Proposition 5 in [3], there is a path P : p 0 , . .…”

Section: Long Rainbow Paths In χ(G)-coloringsmentioning

confidence: 99%

Colorful Paths in Vertex Coloring of Graphs

Akbari

Liaghat

Nikzad

2011

Electron. J. Combin.

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A colorful path in a graph $G$ is a path with $\chi(G)$ vertices whose colors are different. A $v$-colorful path is such a path, starting from $v$. Let $G\neq C_7$ be a connected graph with maximum degree $\Delta(G)$. We show that there exists a $(\Delta(G)+1)$-coloring of $G$ with a $v$-colorful path for every $v\in V(G)$. We also prove that this result is true if one replaces $(\Delta(G)+1)$ colors with $2\chi(G)$ colors. If $\chi(G)=\omega(G)$, then the result still holds for $\chi(G)$ colors. For every graph $G$, we show that there exists a $\chi(G)$-coloring of $G$ with a rainbow path of length $\lfloor\chi(G)/2\rfloor$ starting from each $v \in V(G)$.

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“…Proof. Let c be a χ-coloring of G. As a consequence of Proposition 5 in [3], there is a path P : p 0 , . .…”

Section: Long Rainbow Paths In χ(G)-coloringsmentioning

confidence: 99%

Colorful Paths in Vertex Coloring of Graphs

Akbari

Liaghat

Nikzad

2011

Electron. J. Combin.

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“…This result is mentioned in [3]. We give a sketch of the proof: Orient each edge of G from the vertex of smaller color to the vertex of larger color.…”

Section: A(v) < A(u) < R(u) < R(v)mentioning

confidence: 87%

Locally restricted colorings

Blöchliger

Werra

2006

Discrete Applied Mathematics

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We consider the following problem: given suitable integers and p, what is the smallest value such that, for any graph G with chromatic number and any vertex coloring of G with at most + p colors, there is a vertex v such that at least different colors occur within distance of v? Let ( , p) be this value; we show in particular that ( , p) p/2 + 1 for all , p. We give the exact value of when p = 0 or 3, and ( , p) = (4, 1) or (4, 2).

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