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 Study of Circular Distance in Graphs

2021 ◽  
Vol 12 (2) ◽  
pp. 2437-2444
Author(s):  
Peruri Lakshmi Narayana Varma , Et. al.
Keyword(s):  
Graph Theory ◽  
Significant Role ◽  
Geodesic Distance ◽  
Cartesian Product ◽  
Cartesian Product Of Graphs ◽  
Pendent Vertices ◽  
Product Of Graphs

Circular distance between vertices of a graph has a significant role, which is defined as summation of detour distance and geodesic distance. Attention is paid, this is metric on the set of all vertices of graph  and it plays an important role in graph theory. Some bounds have been carried out for circular distance in terms of pendent vertices of graph  . Some results and properties have been found for circular distance for some classes of graphs and applied this distance to Cartesian product of graphs〖  P〗_2×C_n.  Including 〖 P〗_2×C_n, some graphs acted as a circular self-centered. Using this circular distance there exists some relations between various radii and diameters in path graphs. The possible applications were briefly discussed. 

scholarly journals The b-chromatic number of powers of cycles

2013 ◽  
Vol Vol. 15 no. 1 (Graph Theory) ◽  
Author(s):  
Anja Kohl
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Chromatic Number ◽  
Vertex Coloring ◽  
Color Class ◽  
International Audience ◽  
Proper Vertex

Graph Theory International audience A b-coloring of a graph G by k colors is a proper vertex coloring such that each color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number χb(G) is the maximum integer k for which G has a b-coloring by k colors. Let Cnr be the rth power of a cycle of order n. In 2003, Effantin and Kheddouci established the b-chromatic number χb(Cnr) for all values of n and r, except for 2r+3≤n≤3r. For the missing cases they presented the lower bound L:= min n-r-1,r+1+⌊ n-r-1 / 3⌋ and conjectured that χb(Cnr)=L. In this paper, we determine the exact value on χb(Cnr) for the missing cases. It turns out that χb(Cnr)>L for 2r+3≤n≤2r+3+r-6 / 4.

scholarly journals On (2-d)-kernels in the cartesian product of graphs

2016 ◽  
Vol 70 (2) ◽  
pp. 1 ◽  
Author(s):  
Paweł Bednarz ◽  
Iwona Włoch
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

In this paper we study the problem of the existence of (2-d)-kernels in the cartesian product of graphs. We give sufficient conditions for the existence of (2-d)-kernels in the cartesian product and also we consider the number of (2-d)-kernels.

scholarly journals On very strongly perfect Cartesian product graphs

2021 ◽  
Vol 7 (2) ◽  
pp. 2634-2645
Author(s):  
Ganesh Gandal ◽  
◽  
R Mary Jeya Jothi ◽  
Narayan Phadatare ◽  
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Perfect Graph ◽  
Finite Graphs ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Necessary And Sufficient ◽  
And Control ◽  
Product Of Graphs ◽  
Real Time Application

<abstract><p>Let $ G_1 \square G_2 $ be the Cartesian product of simple, connected and finite graphs $ G_1 $ and $ G_2 $. We give necessary and sufficient conditions for the Cartesian product of graphs to be very strongly perfect. Further, we introduce and characterize the co-strongly perfect graph. The very strongly perfect graph is implemented in the real-time application of a wireless sensor network to optimize the set of master nodes to communicate and control nodes placed in the field.</p></abstract>

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 $\mathcal{S} \in \{ \{ 0,1 \} , \{1,2 \} \}$.

scholarly journals 8-star-choosability of a graph with maximum average degree less than 3

2011 ◽  
Vol Vol. 13 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Min Chen ◽  
André Raspaud ◽  
Weifan Wang
Keyword(s):  
Graph Theory ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Average Degree ◽  
Vertex Coloring ◽  
Maximum Average Degree ◽  
International Audience ◽  
Star Coloring ◽  
List Chromatic Number ◽  
Proper Vertex

Graphs and Algorithms International audience A proper vertex coloring of a graphGis called a star-coloring if there is no path on four vertices assigned to two colors. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring c such that c(v) epsilon L(v). If G is L-star-colorable for any list assignment L with vertical bar L(v)vertical bar \textgreater= k for all v epsilon V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by X-s(l)(G), is the smallest integer k such that G is k-star-choosable. In this article, we prove that every graph G with maximum average degree less than 3 is 8-star-choosable. This extends a result that planar graphs of girth at least 6 are 8-star-choosable [A. Kundgen, C. Timmons, Star coloring planar graphs from small lists, J. Graph Theory, 63(4): 324-337, 2010].

s-strongly perfect cartesian product of graphs

1992 ◽  
Vol 16 (4) ◽  
pp. 297-303
Author(s):  
Elefterie Olaru ◽  
Eugen M??ndrescu
Keyword(s):  
Cartesian Product ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

THE DIAMETER VARIABILITY OF THE CARTESIAN PRODUCT OF GRAPHS

2014 ◽  
Vol 06 (01) ◽  
pp. 1450001 ◽  
Author(s):  
M. R. CHITHRA ◽  
A. VIJAYAKUMAR
Keyword(s):  
Interconnection Networks ◽  
Cartesian Product ◽  
Graph Products ◽  
Single Edge ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.

Sign in / Sign up

Export Citation Format

Share Document