scholarly journals On vertex-coloring edge-weighting of graphs

2009 ◽  
Vol 4 (2) ◽  
pp. 325-334
Author(s):  
Hongliang Lu ◽  
Xu Yang ◽  
Qinglin Yu
Keyword(s):  
Vertex Coloring ◽  
Edge Weighting

scholarly journals Vertex coloring edge-weighting of some wheel related of graphs

2018 ◽  
Author(s):  
Dafik ◽  
R. Alfarisi ◽  
A. I. Kristiana ◽  
R. Adawiyah ◽  
I. H. Agustin
Keyword(s):  
Vertex Coloring ◽  
Edge Weighting

scholarly journals Vertex-coloring 3-edge-weighting of some graphs

2017 ◽  
Vol 340 (2) ◽  
pp. 154-159 ◽  
Author(s):  
Yezhou Wu ◽  
Cun-Quan Zhang ◽  
Bao-Xuan Zhu
Keyword(s):  
Vertex Coloring ◽  
Edge Weighting

scholarly journals Graphs with vertex-coloring and detectable 2-edge-weighting

2016 ◽  
Vol 13 (2) ◽  
pp. 146-156 ◽  
Author(s):  
N. Paramaguru ◽  
R. Sampathkumar
Keyword(s):  
Vertex Coloring ◽  
Edge Weighting

On vertex-coloring 13-edge-weighting

2008 ◽  
Vol 3 (4) ◽  
pp. 581-587 ◽  
Author(s):  
Tao Wang ◽  
Qinglin Yu
Keyword(s):  
Vertex Coloring ◽  
Edge Weighting

scholarly journals On the 1-2-3-conjecture

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Akbar Davoodi ◽  
Behnaz Omoomi
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Bipartite Graphs ◽  
Vertex Coloring ◽  
Cartesian Product Of Graphs ◽  
Edge Weighting ◽  
International Audience ◽  
Product Of Graphs ◽  
Proper Vertex

Graph Theory International audience A k-edge-weighting of a graph G is a function w:E(G)→{1,…,k}. An edge-weighting naturally induces a vertex coloring c, where for every vertex v∈V(G), c(v)=∑e∼vw(e). If the induced coloring c is a proper vertex coloring, then w is called a vertex-coloring k-edge-weighting (VC k-EW). Karoński et al. (J. Combin. Theory Ser. B, 91 (2004) 151 13;157) conjectured that every graph admits a VC 3-EW. This conjecture is known as the 1-2-3-conjecture. In this paper, first, we study the vertex-coloring edge-weighting of the Cartesian product of graphs. We prove that if the 1-2-3-conjecture holds for two graphs G and H, then it also holds for G□H. Also we prove that the Cartesian product of connected bipartite graphs admits a VC 2-EW. Moreover, we present several sufficient conditions for a graph to admit a VC 2-EW. Finally, we explore some bipartite graphs which do not admit a VC 2-EW.

scholarly journals Some unicyclic graphs and its vertex coloring edge-weighting

2018 ◽  
Author(s):  
R. Adawiyah ◽  
Dafik ◽  
I. H. Agustin ◽  
A. I. Kristiana ◽  
R. Alfarisi
Keyword(s):  
Vertex Coloring ◽  
Unicyclic Graphs ◽  
Edge Weighting

scholarly journals Vertex coloring edge-weighting of coronation by path graphs

2019 ◽  
Vol 1211 ◽  
pp. 012004
Author(s):  
A. I. Kristiana ◽  
M. I. Utoyo ◽  
Dafik ◽  
I. H. Agustin ◽  
R. Alfarisi ◽  
...  
Keyword(s):  
Vertex Coloring ◽  
Edge Weighting

scholarly journals Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Hongliang Lu
Keyword(s):  
Bipartite Graph ◽  
Bipartite Graphs ◽  
Vertex Coloring ◽  
Edge Weighting ◽  
International Audience

International audience Let $G$ be a graph and $\mathcal{S}$ be a subset of $Z$. A vertex-coloring $\mathcal{S}$-edge-weighting of $G$ is an assignment of weights by the elements of $\mathcal{S}$ to each edge of $G$ so that adjacent vertices have different sums of incident edges weights. It was proved that every 3-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} = \{1,2 \}$ (H. Lu, Q. Yu and C. Zhang, Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32 (2011), 22-27). In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring $\mathcal{S}$-edge-weighting for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$. These bounds we obtain are tight, since there exists a family of infinite bipartite graphs which are 2-connected and do not admit vertex-coloring $\mathcal{S}$-edge-weightings for $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$.

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