scholarly journals Colour-blind Can Distinguish Colour Pallets

10.37236/3744 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Connected Graph ◽  
Delta G ◽  
Edge Colouring

Let $c:E\to\{1,\ldots,k\}$ be an edge colouring of a connected graph $G=(V,E)$. Each vertex $v$ is endowed with a naturally defined pallet under $c$, understood as the multiset of colours incident with $v$. If $\delta(G)\geq 2$, we obviously (for $k$ large enough) may colour the edges of $G$ so that adjacent vertices are distinguished by their pallets of colours. Suppose then that our coloured graph is examined by a person who is unable to name colours, but perceives if two object placed next to each other are coloured differently. Can we colour $G$ so that this individual can distinguish colour pallets of adjacent vertices? It is proved that if $\delta(G)$ is large enough, then it is possible using just colours 1, 2 and 3. This result is sharp and improves all earlier ones. It also constitutes a strengthening of a result by Addario-Berry, Aldred, Dalal and Reed (2005).

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

scholarly journals A NEIGHBOURHOOD CONDITION FOR GRAPHS TO BE FRACTIONAL (k, m)-DELETED GRAPHS

2009 ◽  
Vol 52 (1) ◽  
pp. 33-40 ◽  
Author(s):  
SIZHONG ZHOU
Keyword(s):  
Connected Graph ◽  
Delta G

AbstractLet G be a connected graph of order n, and let k ≥ 2 and m ≥ 0 be two integers. In this paper, we show that G is a fractional (k, m)-deleted graph if $\delta(G)\,{\geq}\, k+m+\frac{(m+1)^{2}-1}{4k}$, $n\,{\geq}\, 9k-1-4\sqrt{2(k-1)^{2}+2}+2(2k+1)m$ and $|N_G(x)\cup N_G(y)|\,{\geq}\,\frac{1}{2}(n+k-2)$ for each pair of non-adjacent vertices x, y of G. This result is an extension of the previous result of Zhou [11].

scholarly journals Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

10.37236/9039 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Hannah Guggiari ◽  
Alex Scott
Keyword(s):  
Complete Graph ◽  
Minimum Degree ◽  
Connected Component ◽  
Delta G ◽  
Edge Colouring

For every $n\in\mathbb{N}$ and $k\geqslant2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geqslant3$, it is known that the complete graph can be replaced by a graph $G$ with $\delta(G)\geqslant(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show that the maximum possible value of $\varepsilon_3$ is $\frac16$. This disproves a conjecture of Gyárfas and Sárközy.

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 $2t+r$ .

scholarly journals Independent Domination Subdivision in Graphs

2021 ◽  
Author(s):  
Ammar Babikir ◽  
Magda Dettlaff ◽  
Michael A. Henning ◽  
Magdalena Lemańska
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Independent Domination ◽  
Delta G ◽  
Independent Dominating Set ◽  
Domination Subdivision Number

AbstractA set S of vertices in a graph G is a dominating set if every vertex not in S is ad jacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. The independent domination subdivision number $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the independent domination number. We show that for every connected graph G on at least three vertices, the parameter $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is well defined and differs significantly from the well-studied domination subdivision number $$\mathrm{sd_\gamma }(G)$$ sd γ ( G ) . For example, if G is a block graph, then $$\mathrm{sd_\gamma }(G) \le 3$$ sd γ ( G ) ≤ 3 , while $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) can be arbitrary large. Further we show that there exist connected graph G with arbitrarily large maximum degree $$\Delta (G)$$ Δ ( G ) such that $$ \hbox {sd}_{\mathrm{i}}(G) \ge 3 \Delta (G) - 2$$ sd i ( G ) ≥ 3 Δ ( G ) - 2 , in contrast to the known result that $$\mathrm{sd_\gamma }(G) \le 2 \Delta (G) - 1$$ sd γ ( G ) ≤ 2 Δ ( G ) - 1 always holds. Among other results, we present a simple characterization of trees T with $$ \hbox {sd}_{\mathrm{i}}(T) = 1$$ sd i ( T ) = 1 .

scholarly journals Spanning Trails in a 2-Connected Graph

10.37236/8121 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Shipeng Wang ◽  
Liming Xiong
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Free Graph ◽  
Delta G ◽  
Exceptional Families

In this article we prove the following: Let $G$ be a $2$-connected graph with circumference $c(G)$. If  $c(G)\leq 5$, then $G$ has a spanning trail starting from any vertex, if  $c(G)\leq 7$, then $G$ has a spanning trail.  As applications of  this result, we obtain the following. Every $2$-edge-connected graph of order at most 8 has a spanning trail starting from any vertex  with the exception of six graphs.  Let $G$ be a $2$-edge-connected graph and $S$ a subset of $V(G)$ such that $E(G-S)=\emptyset$ and $|S|\leq 6$. Then $G$ has a trail traversing all vertices of $S$ with the exception of two graphs, moreover, if $|S|\leq 4$, then $G$ has a trail starting from any vertex of $S$ and containing $S$. Every $2$-connected claw-free graph $G$ with order $n$ and minimum degree $\delta(G)> \frac{n}{7}+4\geq 23$ is traceable or belongs to two exceptional families of well-defined  graphs, and moreover, if $\delta(G)> \frac{n}{6}+4\geq 13$, then $G$ is traceable. All above results are sharp in a sense.

scholarly journals The Chromatic Index of a Graph Whose Core has Maximum Degree $2$

10.37236/2101 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Mikio Kano ◽  
Saieed Akbari ◽  
Maryam Ghanbari ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Chromatic Index ◽  
The Core ◽  
Class 1 ◽  
Minimum Number ◽  
Even Order ◽  
Delta G

Let $G$ be a graph. The core of $G$, denoted by $G_{\Delta}$, is the subgraph of $G$ induced by the vertices of degree $\Delta(G)$, where $\Delta(G)$ denotes the maximum degree of $G$. A $k$-edge coloring of $G$ is a function $f:E(G)\rightarrow L$ such that $|L| = k$ and $f(e_1)\neq f(e_2)$ for all two adjacent edges  $e_1$ and $e_2$ of $G$. The chromatic index of $G$, denoted by $\chi'(G)$, is the minimum number $k$ for which $G$ has a $k$-edge coloring.  A graph $G$ is said to be Class $1$ if $\chi'(G) = \Delta(G)$ and Class $2$ if $\chi'(G) = \Delta(G) + 1$. In this paper it is shown that every connected graph $G$ of even order and with $\Delta(G_{\Delta})\leq 2$ is Class $1$ if $|G_{\Delta}|\leq 9$ or $G_{\Delta}$ is a cycle of order $10$.

scholarly journals The Distinguishing Index of Infinite Graphs

10.37236/3933 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Izak Broere ◽  
Monika Pilśniak
Keyword(s):  
Random Graph ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Distinguishing Number ◽  
General Bound ◽  
Delta G ◽  
Edge Colouring

The  distinguishing index $D^\prime(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has an edge colouring with $d$ colours that is only preserved by the trivial automorphism. This is similar to the notion of the distinguishing number $D(G)$ of a graph $G$, which is defined with respect to vertex colourings.We derive several bounds for infinite graphs, in particular, we prove the general bound $D^\prime(G)\leq\Delta(G)$ for an arbitrary infinite graph. Nonetheless,  the distinguishing index is at most two for many countable graphs, also for the infinite random graph and for uncountable tree-like graphs.We also investigate the concept of the motion of edges and its relationship with the Infinite Motion Lemma. 

scholarly journals Variations on the Petersen Colouring Conjecture

10.37236/8515 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
François Pirot ◽  
Jean-Sébastien Sereni ◽  
Riste Škrekovski
Keyword(s):  
Connected Graph ◽  
Perfect Matching ◽  
Petersen Graph ◽  
Cubic Graph ◽  
Edge Colouring

The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with 5 colours such that for every edge e, the set of colours assigned to the edges adjacent to e has cardinality either 2 or 4, but not 3. We prove that every bridgeless cubic graph $G$ admits an edge-colouring with 4 colours such that at most $8/15\cdot|E(G)|$ edges do not satisfy the above condition. This bound is tight and the Petersen graph is the only connected graph for which the bound cannot be decreased. We obtain such a 4-edge-colouring by using a carefully chosen subset of edges of a perfect matching, and the analysis relies on a simple discharging procedure with essentially no reductions and very few rules.

scholarly journals Strong Menger Connectedness of Augmented k-ary n-cubes

2020 ◽  
Author(s):  
Mei-Mei Gu ◽  
Jou-Ming Chang ◽  
Rong-Xia Hao
Keyword(s):  
Connected Graph ◽  
Fault Tolerant ◽  
Disjoint Paths ◽  
Edge Connectivity ◽  
Arbitrary Vertex ◽  
Edge Disjoint ◽  
Vertex Set ◽  
Delta G ◽  
Disjoint Edge

Abstract A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{\textrm{deg}_G(x), \textrm{deg}_G(y)\}$ internally disjoint (edge disjoint) paths between $x$ and $y$. Motivated by parallel routing in networks with faults, Oh and Chen (resp., Qiao and Yang) proposed the (fault-tolerant) strong Menger (edge) connectivity as follows. A graph $G$ is called $m$-strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$. A graph $G$ is called $m$-conditional strongly Menger (edge) connected if $G-F$ remains strongly Menger (edge) connected for an arbitrary vertex set $F\subseteq V(G)$ (resp. edge set $F\subseteq E(G)$) with $|F|\leq m$ and $\delta (G-F)\geq 2$. In this paper, we consider strong Menger (edge) connectedness of the augmented $k$-ary $n$-cube $AQ_{n,k}$, which is a variant of $k$-ary $n$-cube $Q_n^k$. By exploring the topological proprieties of $AQ_{n,k}$, we show that $AQ_{n,3}$ (resp. $AQ_{n,k}$, $k\geq 4$) is $(4n-9)$-strongly (resp. $(4n-8)$-strongly) Menger connected for $n\geq 4$ (resp. $n\geq 2$) and $AQ_{n,k}$ is $(4n-4)$-strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. Moreover, we obtain that $AQ_{n,k}$ is $(8n-10)$-conditional strongly Menger edge connected for $n\geq 2$ and $k\geq 3$. These results are all optimal in the sense of the maximum number of tolerated vertex (resp. edge) faults.

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