scholarly journals On the Size of $(K_t,\mathcal{T}_k)$-Co-Critical Graphs

10.37236/8857 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Zi-Xia Song ◽  
Jingmei Zhang
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Minimum Degree ◽  
Bootstrap Percolation ◽  
Critical Graphs ◽  
Critical Graph ◽  
The Family ◽  
Saturated Graphs ◽  
Minimum Number ◽  
Monochromatic Copy

Given an integer $r\geqslant 1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A non-complete graph $G$ is $(H_1, \ldots, H_r)$-co-critical if $G \nrightarrow ({H}_1, \ldots, {H}_r)$, but $G+e\rightarrow ({H}_1, \ldots, {H}_r)$ for every edge $e$ in $\overline{G}$. In this paper, motivated by Hanson and Toft's conjecture [Edge-colored saturated graphs, J Graph Theory 11(1987), 191–196], we study the minimum number of edges over all $(K_t, \mathcal{T}_k)$-co-critical graphs on $n$ vertices, where $\mathcal{T}_k$ denotes the family of all trees on $k$ vertices. Following Day [Saturated graphs of prescribed minimum degree, Combin. Probab. Comput. 26 (2017), 201–207], we apply graph bootstrap percolation on a not necessarily $K_t$-saturated graph to prove that for all $t\geqslant4 $ and $k\geqslant\max\{6, t\}$, there exists a constant $c(t, k)$ such that, for all $n \ge (t-1)(k-1)+1$, if $G$ is a $(K_t, \mathcal{T}_k)$-co-critical graph on $n$ vertices, then $$ e(G)\geqslant \left(\frac{4t-9}{2}+\frac{1}{2}\left\lceil \frac{k}{2} \right\rceil\right)n-c(t, k).$$ Furthermore, this linear bound is asymptotically best possible when $t\in\{4,5\}$ and $k\geqslant6$. The method we develop in this paper may shed some light on attacking Hanson and Toft's conjecture.

On Dirac's Generalization of Brooks' Theorem

1972 ◽  
Vol 24 (5) ◽  
pp. 805-807 ◽  
Author(s):  
Hudson V. Kronk ◽  
John Mitchem
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Equivalent Formulation ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Cycle ◽  
Proper Subgraph

It is easy to verify that any connected graph G with maximum degree s has chromatic number χ(G) ≦ 1 + s. In [1], R. L. Brooks proved that χ(G) ≦ s, unless s = 2 and G is an odd cycle or s > 2 and G is the complete graph Ks+1. This was the first significant theorem connecting the structure of a graph with its chromatic number. For s ≦ 4, Brooks' theorem says that every connected s-chromatic graph other than Ks contains a vertex of degree > s — 1. An equivalent formulation can be given in terms of s-critical graphs. A graph G is said to be s-critical if χ(G) = s, but every proper subgraph has chromatic number less than s. Each scritical graph has minimum degree ≦ s — 1. We can now restate Brooks' theorem: if an s-critical graph, s ≦ 4, is not Ks and has p vertices and q edges, then 2q ≦ (s — l)p + 1. Dirac [2] significantly generalized the theorem of Brooks by showing that 2q ≦ (s — 1)£ + s — 3 and that this result is best possible.

Monochromatic paths and cycles in 2-edge-coloured graphs with large minimum degree

2021 ◽  
pp. 1-14
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Mikhail Lavrov ◽  
Xujun Liu
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Dense Subgraph ◽  
Degree Condition ◽  
Vertex Graph ◽  
Delta G ◽  
Edge Colouring ◽  
Monochromatic Copy

Abstract A graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$ , then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour. Our result is tight in the sense that no longer cycles (of length $>2t+r$ ) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$ -vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$ , in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$ .

Ramsey Problems with Bounded Degree Spread

1993 ◽  
Vol 2 (3) ◽  
pp. 263-269 ◽  
Author(s):  
G. Chen ◽  
R. H. Schelp
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Bipartite Graphs ◽  
Bounded Degree ◽  
Monochromatic Copy

Let k be a positive integer, k ≥ 2. In this paper we study bipartite graphs G such that, for n sufficiently large, each two-coloring of the edges of the complete graph Kn gives a monochromatic copy of G, with some k of its vertices having the maximum degree of these k vertices minus the minimum degree of these k vertices (in the colored Kn) at most k − 2.

MATCHING PROPERTIES IN DOUBLE DOMINATION EDGE CRITICAL GRAPHS

2010 ◽  
Vol 02 (02) ◽  
pp. 151-160 ◽  
Author(s):  
HAICHAO WANG ◽  
LIYING KANG
Keyword(s):  
Perfect Matching ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Order ◽  
Even Order ◽  
Double Domination

A vertex subset S of a graph G = (V, E) is a double dominating set for G if |N[v]∩S| ≥ 2 for each vertex v ∈ V, where N[v] = {u |uv ∈ E}∪{v}. The double domination number of G, denoted by γ×2(G), is the cardinality of a smallest double dominating set of G. A graph G is said to be double domination edge critical if γ×2(G + e) < γ×2(G) for any edge e ∉ E. A double domination edge critical graph G with γ×2(G) = k is called k - γ×2(G)-critical. In this paper, we first show that G has a perfect matching if G is a connected 3 - γ×2(G)-critical graph of even order. Secondly, we show that G is factor-critical if G is a connected 3 - γ×2(G)-critical graph with odd order and minimum degree at least 2. Finally, we show that G is factor-critical if G is a connected K1,4-free 4 - γ×2(G)-critical graph of odd order with minimum degree at least 2.

scholarly journals Ramsey Numbers of Ordered Graphs

10.37236/7816 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Martin Balko ◽  
Josef Cibulka ◽  
Karel Král ◽  
Jan Kynčl
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Positive Answer ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Minimum Number ◽  
Ordered Graphs ◽  
Constant Interval ◽  
Monochromatic Copy ◽  
Special Classes

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

scholarly journals Structural Properties of Connected Domination Critical Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2568
Author(s):  
Norah Almalki ◽  
Pawaton Kaemawichanurat
Keyword(s):  
Structural Properties ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Domination Number ◽  
Free Graph ◽  
Critical Graphs ◽  
Critical Graph ◽  
Odd Order ◽  
Connected Domination Number ◽  
Even Order

A graph G is said to be k-γc-critical if the connected domination number γc(G) is equal to k and γc(G+uv)<k for any pair of non-adjacent vertices u and v of G. Let ζ be the number of cut vertices of G and let ζ0 be the maximum number of cut vertices that can be contained in one block. For an integer ℓ≥0, a graph G is ℓ-factor critical if G−S has a perfect matching for any subset S of vertices of size ℓ. It was proved by Ananchuen in 2007 for k=3, Kaemawichanurat and Ananchuen in 2010 for k=4 and by Kaemawichanurat and Ananchuen in 2020 for k≥5 that every k-γc-critical graph has at most k−2 cut vertices and the graphs with maximum number of cut vertices were characterized. In 2020, Kaemawichanurat and Ananchuen proved further that, for k≥4, every k-γc-critical graphs satisfies the inequality ζ0(G)≤mink+23,ζ. In this paper, we characterize all k-γc-critical graphs having k−3 cut vertices. Further, we establish realizability that, for given k≥4, 2≤ζ≤k−2 and 2≤ζ0≤mink+23,ζ, there exists a k-γc-critical graph with ζ cut vertices having a block which contains ζ0 cut vertices. Finally, we proved that every k-γc-critical graph of odd order with minimum degree two is 1-factor critical if and only if 1≤k≤2. Further, we proved that every k-γc-critical K1,3-free graph of even order with minimum degree three is 2-factor critical if and only if 1≤k≤2.

scholarly journals Saturated Graphs of Prescribed Minimum Degree

2016 ◽  
Vol 26 (2) ◽  
pp. 201-207 ◽  
Author(s):  
A. NICHOLAS DAY
Keyword(s):  
Minimum Degree ◽  
Upper Bounds ◽  
Saturated Graphs ◽  
Minimum Number

A graph G is H-saturated if it contains no copy of H as a subgraph but the addition of any new edge to G creates a copy of H. In this paper we are interested in the function satt(n,p), defined to be the minimum number of edges that a Kp-saturated graph on n vertices can have if it has minimum degree at least t. We prove that satt(n,p) = tn − O(1), where the limit is taken as n tends to infinity. This confirms a conjecture of Bollobás when p = 3. We also present constructions for graphs that give new upper bounds for satt(n,p).

A maximum degree theorem for diameter-2-critical graphs

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Teresa Haynes ◽  
Michael Henning ◽  
Lucas Merwe ◽  
Anders Yeo
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Discrete Math ◽  
Bipartite Graphs ◽  
Extremal Graphs ◽  
Critical Graphs ◽  
Critical Graph ◽  
Complete Bipartite Graphs

AbstractA graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ⌊n 2/4⌋ and that the extremal graphs are the complete bipartite graphs K ⌊n/2⌋,⌊n/2⌉. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n 0 where n 0 is a tower of 2’s of height about 1014. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.

scholarly journals Fractional Matchings, Component-Factors and Edge-Chromatic Critical Graphs

2021 ◽  
Author(s):  
Antje Klopp ◽  
Eckhard Steffen
Keyword(s):  
Upper Bounds ◽  
Matching Number ◽  
Critical Graphs ◽  
Critical Graph ◽  
Minimum Number

AbstractThe first part of the paper studies star-cycle factors of graphs. It characterizes star-cycle factors of a graph G and proves upper bounds for the minimum number of $$K_{1,2}$$ K 1 , 2 -components in a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor of a graph G. Furthermore, it shows where these components are located with respect to the Gallai–Edmonds decomposition of G and it characterizes the edges which are not contained in any $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor of G. The second part of the paper proves that every edge-chromatic critical graph G has a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor, and the number of $$K_{1,2}$$ K 1 , 2 -components is bounded in terms of its fractional matching number. Furthermore, it shows that for every edge e of G, there is a $$\{K_{1,1}, K_{1,2}, C_n:n\ge 3\}$$ { K 1 , 1 , K 1 , 2 , C n : n ≥ 3 } -factor F with $$e \in E(F)$$ e ∈ E ( F ) . Consequences of these results for Vizing’s critical graph conjectures are discussed.

scholarly journals New Upper Bounds for the Heights of Some Light Subgraphs in 1-Planar Graphs with High Minimum Degree

2011 ◽  
Vol Vol. 13 no. 3 (Combinatorics) ◽  
Author(s):  
Xin Zhang ◽  
Jian-Liang Wu ◽  
Guizhen Liu
Keyword(s):  
Planar Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Upper Bounds ◽  
The Family ◽  
International Audience

Combinatorics International audience A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph of minimum degree 6 contains a copy of 4-cycle with all vertices of degree at most 19. In addition, we also show that the complete graph K 4 is light in the family of 1-planar graphs of minimum degree 7, with its height at most 11.

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