scholarly journals A further result on the potential-Ramsey number of G1 and G2

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1605-1617
Author(s):  
Jinzhi Du ◽  
Jianhua Yin
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Degree Sequence ◽  
Simple Graph ◽  
Negative Integer ◽  
Ramsey Number ◽  
Complementary Sequence ◽  
Split Graph ◽  
Graphic Sequence

A non-increasing sequence ? = (d1,. . ., dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of ?. Given a graph H, a graphic sequence ? is potentially H-graphic if ? has a realization containing H as a subgraph. Busch et al. (Graphs Combin., 30(2014)847-859) considered a degree sequence analogue to classical graph Ramsey number as follows: for graphs G1 and G2, the potential-Ramsey number rpot(G1,G2) is the smallest non-negative integer k such that for any k-term graphic sequence ?, either ? is potentially G1-graphic or the complementary sequence ? = (k - 1 - dk,..., k - 1 - d1) is potentially G2-graphic. They also gave a lower bound on rpot(G;Kr+1) for a number of choices of G and determined the exact values for rpot(Kn;Kr+1), rpot(Cn;Kr+1) and rpot(Pn,Kr+1). In this paper, we will extend the complete graph Kr+1 to the complete split graph Sr,s = Kr ? Ks. Clearly, Sr,1 = Kr+1. We first give a lower bound on rpot(G, Sr,s) for a number of choices of G, and then determine the exact values for rpot(Cn, Sr,s) and rpot(Pn, Sr,s).

scholarly journals HAMILTONICITY OF CAMOUFLAGE GRAPHS

2021 ◽  
Vol 5 (10) ◽  
Author(s):  
Sowmiya K
Keyword(s):  
Planar Graphs ◽  
Degree Sequence ◽  
Simple Graph ◽  
Hamilton Path ◽  
Graphic Sequence ◽  
Vertex Transitive ◽  
Connected Vertex

This paper examines the Hamiltonicity of graphs having some hidden behaviours of some other graphs in it. The well-known mathematician Barnette introduced the open conjecture which becomes a theorem by restricting our attention to the class of graphs which is 3-regular, 3- connected, bipartite, planar graphs having odd number of vertices in its partition be proved as a Hamiltonian. Consequently the result proved in this paper stated that “Every connected vertex-transitive simple graph has a Hamilton path” shows a significant improvement over the previous efforts by L.Babai and L.Lovasz who put forth this conjecture. And we characterize a graphic sequence which is forcibly Hamiltonian if every simple graph with degree sequence is Hamiltonian. Thus we discussed about the concealed graphs which are proven to be Hamiltonian.

scholarly journals An extremal problem for a graphic sequence to have a realization containing every 2-tree with prescribed size

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
De-Yan Zeng ◽  
Jian-Hua Yin
Keyword(s):  
Lower Bound ◽  
Extremal Problem ◽  
Simple Graph ◽  
Graphic Sequence ◽  
International Audience

International audience A graph $G$ is a $2$<i>-tree</i> if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is a 2-tree. Clearly, if $G$ is a 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence $\pi =(d_1, \ldots ,d_n)$ of nonnegative integers is a <i>graphic sequence</i> if it is realizable by a simple graph $G$ on $n$ vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795&#x2013;802) proved that if $k \geq 2$, $n \geq \frac{9}{2}k^2 + \frac{19}{2}k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > (k-2)n$, then $\pi$ has a realization containing every tree on $k$ vertices as a subgraph. Moreover, the lower bound $(k-2)n$ is the best possible. This is a variation of a conjecture due to Erd&#x0151;s and S&oacute;s. In this paper, we investigate an analogue extremal problem for 2-trees and prove that if $k \geq 3$, $n \geq 2k^2-k$ and $\pi =(d_1, \ldots ,d_n)$ is a graphic sequence with $\sum \limits_{i=1}^n d_i > \frac{4kn}{3} - \frac{5n}{3}$ then $\pi$ has a realization containing every 2-tree on $k$ vertices as a subgraph. We also show that the lower bound $\frac{4kn}{3} - \frac{5n}{3}$ is almost the best possible.

scholarly journals A Note on Edge-Colourings Avoiding Rainbow $K_4$ and Monochromatic $K_m$

10.37236/257 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Veselin Jungić ◽  
Tomáš Kaiser ◽  
Daniel Král'
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Projective Planes ◽  
Ramsey Number ◽  
Complete Subgraph ◽  
Finite Projective Planes ◽  
Edge Colourings ◽  
Edge Colouring

We study the mixed Ramsey number $maxR(n,{K_m},{K_r})$, defined as the maximum number of colours in an edge-colouring of the complete graph $K_n$, such that $K_n$ has no monochromatic complete subgraph on $m$ vertices and no rainbow complete subgraph on $r$ vertices. Improving an upper bound of Axenovich and Iverson, we show that $maxR(n,{K_m},{K_4}) \leq n^{3/2}\sqrt{2m}$ for all $m\geq 3$. Further, we discuss a possible way to improve their lower bound on $maxR(n,{K_4},{K_4})$ based on incidence graphs of finite projective planes.

PARTIAL RESULT ON HADWIGER'S CONJECTURE

2010 ◽  
Vol 02 (03) ◽  
pp. 413-423 ◽  
Author(s):  
ZI-XIA SONG
Keyword(s):  
Degree Sequence ◽  
Simple Graph ◽  
Partial Result ◽  
Degree Sequences ◽  
Induced Subgraph ◽  
Graphic Sequence ◽  
Hadwiger's Conjecture ◽  
Hadwiger’S Conjecture

Let D = (d1, d2, …, dn) be a graphic sequence with 0 ≤ d1 ≤ d2 ≤ ⋯ ≤ dn. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. We say that D is H-free if no graph in R[D] contains H as an induced subgraph. In this paper, we prove that Hadwiger's Conjecture is true for graphs whose degree sequences are claw-free or [Formula: see text]-free.

Extremal graph on normalized Laplacian spectral radius and energy

2015 ◽  
Vol 29 ◽  
pp. 237-253 ◽  
Author(s):  
Kinkar Das ◽  
SHAOWEI SUN
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Complete Graph ◽  
Connected Graph ◽  
Simple Graph ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Isolated Vertex

Let $G=(V,\,E)$ be a simple graph of order $n$ and the normalized Laplacian eigenvalues $\rho_1\geq \rho_2\geq \cdots\geq\rho_{n-1}\geq \rho_n=0$. The normalized Laplacian energy (or Randi\'c energy) of $G$ without any isolated vertex is defined as $$RE(G)=\sum_{i=1}^{n}|\rho_i-1|.$$ In this paper, a lower bound on $\rho_1$ of connected graph $G$ ($G$ is not isomorphic to complete graph) is given and the extremal graphs (that is, the second minimal normalized Laplacian spectral radius of connected graphs) are characterized. Moreover, Nordhaus-Gaddum type results for $\rho_1$ are obtained. Recently, Gutman et al.~gave a conjecture on Randi\'c energy of connected graph [I. Gutman, B. Furtula, \c{S}. B. Bozkurt, On Randi\'c energy, Linear Algebra Appl. 442 (2014) 50--57]. Here this conjecture for starlike trees is proven.

scholarly journals On Multicolor Ramsey Number of Paths Versus Cycles

10.37236/511 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Gholam Reza Omidi ◽  
Ghaffar Raeisi
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Complete Graph ◽  
Ramsey Number ◽  
Subgraph Isomorphic ◽  
Multicolor Ramsey Number

Let $G_1, G_2, G_3, \ldots , G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, \ldots, G_t)$ is the smallest positive integer $n$ such that if the edges of a complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs $H_1,H_2,\ldots,H_t$, then at least one $H_i$ has a subgraph isomorphic to $G_i$. In this paper, we provide the exact value of $R(P_{n_1}, P_{n_2},\ldots, P_{n_t},C_k)$ for certain values of $n_i$ and $k$. In addition, the exact values of $R(P_5,C_4,P_k)$, $R(P_4,C_4,P_k)$, $R(P_5,P_5,P_k)$ and $R(P_5,P_6,P_k)$ are given. Finally, we give a lower bound for $R(P_{2n_1}, P_{2n_2},\ldots, P_{2n_t})$ and we conjecture that this lower bound is the exact value of this number. Moreover, some evidence is given for this conjecture.

scholarly journals On multigraphic and potentially multigraphic sequences

2017 ◽  
Vol 9 (1) ◽  
pp. 35-47 ◽  
Author(s):  
Shariefuddin Pirzada ◽  
Bilal Ahmad Chat ◽  
Uma Tul Samee
Keyword(s):  
Complete Graph ◽  
Split Graph ◽  
Graphic Sequence ◽  
Split Graphs

AbstractAn r-graph(or a multigraph) is a loopless graph in which no two vertices are joined by more than r edges. An r-complete graph on n vertices, denoted by Kn(r), is an r-graph on n vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π = (d1,d2,..., dn) of non-negative integers is said to be r-graphic if it is realizable by an r-graph on n vertices. An r-graphic sequence π is said to be potentially SL;M(r)-graphic if it has a realization containing SL;M(r)as a subgraph. We obtain conditions for an r-graphic sequence to be potentially S(r) L;M-graphic. These are generalizations from split graphs to p-tuple r-split graph.

scholarly journals Asymptotic Bounds on Graphical Partitions and Partition Comparability

2020 ◽  
Author(s):  
Stephen Melczer ◽  
Marcus Michelen ◽  
Somabha Mukherjee
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Degree Sequence ◽  
Simple Graph ◽  
Dominance Order ◽  
Integer Partition ◽  
Asymptotic Bounds

Abstract An integer partition is called graphical if it is the degree sequence of a simple graph. We prove that the probability that a uniformly chosen partition of size $n$ is graphical decreases to zero faster than $n^{-.003}$, answering a question of Pittel. A lower bound of $n^{-1/2}$ was proven by Erd̋s and Richmond, meaning our work demonstrates that the probability decreases polynomially. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.

scholarly journals New Lower Bound for Multicolor Ramsey Numbers for Even Cycles

10.37236/1980 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Tomasz Dzido ◽  
Andrzej Nowik ◽  
Piotr Szuca
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Finite Family ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Multicolor Ramsey Number ◽  
Monochromatic Copy

For given finite family of graphs $G_{1}, G_{2}, \ldots , G_{k}, k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, \ldots , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph on $n$ vertices with $k$ colors then there is always a monochromatic copy of $G_{i}$ colored with $i$, for some $1 \leq i \leq k$. We give a lower bound for $k-$color Ramsey number $R(C_{m}, C_{m}, \ldots , C_{m})$, where $m \geq 4$ is even and $C_{m}$ is the cycle on $m$ vertices.

scholarly journals Ramsey Numbers for Theta Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
M. M. M. Jaradat ◽  
M. S. A. Bataineh ◽  
S. M. E. Radaideh
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Subgraph Isomorphic

The graph Ramsey number is the smallest integer with the property that any complete graph of at least vertices whose edges are colored with two colors (say, red and blue) contains either a subgraph isomorphic to all of whose edges are red or a subgraph isomorphic to all of whose edges are blue. In this paper, we consider the Ramsey numbers for theta graphs. We determine , for . More specifically, we establish that for . Furthermore, we determine for . In fact, we establish that if is even, if is odd.

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