Colorful Paths in Vertex Coloring of Graphs

Akbari, S.; Liaghat, Vahid; Nikzad, Afshin

doi:10.37236/504

Cited by 18 publications

(27 citation statements)

References 4 publications

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“…As observed in [4], the conjecture of Akbari, Khaghanpoor, and Moazzeni [1] holds for G if D f contains a directed cycle. Therefore, we may assume that D f is an acyclic digraph for every k-coloring f of G.…”

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confidence: 62%

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Optimal Colorings with Rainbow Paths

Bendele

Rautenbach

2017

Graphs and Combinatorics

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Let G be a connected graph of chromatic number k. For a k-coloring f of G, a full f -rainbow path is a path of order k in G whose vertices are all colored differently by f .We show that G has a k-coloring f such that every vertex of G lies on a full f -rainbow path, which provides a positive answer to a question posed by Lin (Simple proofs of results on paths representing all colors in proper vertex-colorings, Graphs Combin. 23 (2007) 201-203). Furthermore, we show that if G has a cycle of length k, then G has a k-coloring f such that, for every vertex u of G, some full f -rainbow path begins at u, which solves a problem posed by Bessy and Bousquet (Colorful paths for 3-chromatic graphs, arXiv 1503.00965v1). Finally, we establish some more results on the existence of optimal colorings with (directed) full rainbow paths.

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confidence: 62%

“…Iteratively applying this shifting operation, we may assume that |c −1 (I)| |V (G)| ≥ |I| n . Now, Theorem 2 is a consequence of the observations following (1) and (2) in the proof of Theorem 1. ✷ Proof of Theorem 3: Our proof relies on arguments from [2,4].…”

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confidence: 74%

Optimal Colorings with Rainbow Paths

Bendele

Rautenbach

2017

Graphs and Combinatorics

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“…, k. In particular, if G is connected, then each vertex is the starting vertex of a path containing all colors, as also proved in [6,7]. [1,2,3,6,7] study k-colorings with long rainbow paths starting with any prescribed vertex. In this note we apply the circular chromatic number to refine some of those results.…”

Section: Introductionmentioning

confidence: 87%

“…We begin by extending an elegant coloring lemma by Akbari, Liaghat, and Nikzad [2] to circular colorings.…”

Section: Application Of Circular Coloring To Rainbow Pathsmentioning

confidence: 99%

Rainbow Paths with Prescribed Ends

Alishahi¹,

Taherkhani²,

Thomassen³

2011

Electron. J. Combin.

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It was conjectured in [S. Akbari, F. Khaghanpoor, and S. Moazzeni. Colorful paths in vertex coloring of graphs. Preprint] that, if $G$ is a connected graph distinct from $C_7$, then there is a $\chi(G)$-coloring of $G$ in which every vertex $v\in V(G)$ is an initial vertex of a path $P$ with $\chi(G)$ vertices whose colors are different. In [S. Akbari, V. Liaghat, and A. Nikzad. Colorful paths in vertex coloring of graphs. Electron. J. Combin. 18(1): P17, 9pp, 2011] this was proved with $\lfloor\frac{\chi(G)}{2} \rfloor $ vertices instead of $\chi(G)$ vertices. We strengthen this to $\chi(G)-1$ vertices. We also prove that every connected graph with at least one edge has a proper $k$-coloring (for some $k$) such that every vertex of color $i$ has a neighbor of color $i+1$ (mod $k$). $C_5$ shows that $k$ may have to be greater than the chromatic number. However, if the graph is connected, infinite and locally finite, and has finite chromatic number, then the $k$-coloring exists for every $k \geq \chi(G)$. In fact, the $k$-coloring can be chosen such that every vertex is a starting vertex of an infinite path such that the color increases by $1$ (mod $k$) along each edge. The method is based on the circular chromatic number $\chi_c(G)$. In particular, we verify the above conjecture for all connected graphs whose circular chromatic number equals the chromatic number.

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“…In the following, we will consider modifications of colors and all these modifications have to be understood modulo χ. As defined in [3], the oriented graph D c has vertex set V and ab is an arc of D c if {a, b} is an edge of G and the color of b equals the color of a plus one (this oriented graph was first introduced in [9,13]). A colorful path starting at the vertex x is called a certifying path for…”

Section: Preliminariesmentioning

confidence: 99%