Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs

2010 ◽  
Vol 27 (1) ◽  
pp. 121-128 ◽  
Author(s):  
Tao Jiang ◽  
Michael Salerno
Keyword(s):  
Bipartite Graphs ◽  
Complete Graphs ◽  
Ramsey Numbers

scholarly journals On randomly 3-axial graphs

1982 ◽  
Vol 25 (2) ◽  
pp. 187-206
Author(s):  
Yousef Alavi ◽  
Sabra S. Anderson ◽  
Gary Chartrand ◽  
S.F. Kapoor
Keyword(s):  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Complete Graphs ◽  
Initial Vertex ◽  
Complete Bipartite Graphs

A graph G, every vertex of which has degree at least three, is randomly 3-axial if for each vertex v of G, any ordered collection of three paths in G of length one with initial vertex v can be cyclically randomly extended to produce three internally disjoint paths which contain all the vertices of G. Randomly 3-axial graphs of order p > 4 are characterized for p ≢ 1 (mod 3), and are shown to be either complete graphs or certain regular complete bipartite graphs.

Ramsey numbers of large books and bipartite graphs with small bandwidth

2021 ◽  
Vol 344 (7) ◽  
pp. 112427
Author(s):  
Chunlin You ◽  
Qizhong Lin ◽  
Xun Chen
Keyword(s):  
Bipartite Graphs ◽  
Ramsey Numbers ◽  
Small Bandwidth

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

scholarly journals All partitions have small parts — Gallai–Ramsey numbers of bipartite graphs

2019 ◽  
Vol 254 ◽  
pp. 196-203 ◽  
Author(s):  
Haibo Wu ◽  
Colton Magnant ◽  
Pouria Salehi Nowbandegani ◽  
Suman Xia
Keyword(s):  
Bipartite Graphs ◽  
Ramsey Numbers ◽  
Small Parts

scholarly journals Can Colour-Blind Distinguish Colour Palettes?

10.37236/2319 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Rafał Kalinowski ◽  
Monika Pilśniak ◽  
Jakub Przybyło ◽  
Mariusz Woźniak
Keyword(s):  
Large Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Regular Graphs ◽  
Lovász Local Lemma ◽  
Local Lemma ◽  
Comprehensive Information ◽  
Lovasz Local Lemma ◽  
Delta G ◽  
Irregular Graphs

Let $c:E(G)\rightarrow [k]$ be  a colouring, not necessarily proper, of edges of a graph $G$. For a vertex $v\in V$, let $\overline{c}(v)=(a_1,\ldots,a_k)$, where $ a_i =|\{u:uv\in E(G),\;c(uv)=i\}|$, for $i\in [k].$ If we re-order the sequence $\overline{c}(v)$ non-decreasingly, we obtain a sequence $c^*(v)=(d_1,\ldots,d_k)$, called a palette of a vertex $v$. This can be viewed as the most comprehensive information about colours incident with $v$ which can be delivered by a person who is unable to name colours but distinguishes one from another. The smallest $k$ such that $c^*$ is a proper colouring of vertices of $G$ is called the colour-blind index of a graph $G$, and is denoted by dal$(G)$. We conjecture that there is a constant $K$ such that dal$(G)\leq K$ for every graph $G$ for which the parameter is well defined. As our main result we prove that $K\leq 6$ for regular graphs of sufficiently large degree, and for irregular graphs with $\delta (G)$ and $\Delta(G)$ satisfying certain conditions. The proofs are based on the Lopsided Lovász Local Lemma. We also show that $K=3$ for all regular bipartite graphs, and for complete graphs of order $n\geq 8$.

scholarly journals Consecutive Colouring of Oriented Graphs

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Marta Borowiecka-Olszewska ◽  
Ewa Drgas-Burchardt ◽  
Nahid Yelene Javier-Nol ◽  
Rita Zuazua
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Oriented Graphs ◽  
Complete Problem ◽  
Np Complete

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

Long Paths in Bipartite Graphs and Path-Bistar Bipartite Ramsey Numbers

2019 ◽  
Vol 36 (1) ◽  
pp. 167-176
Author(s):  
Michitaka Furuya ◽  
Shun-ichi Maezawa ◽  
Kenta Ozeki
Keyword(s):  
Bipartite Graphs ◽  
Ramsey Numbers ◽  
Long Paths
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