scholarly journals The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

2006 ◽  
Vol 56 (2) ◽  
pp. 317-338 ◽  
Author(s):  
Ladislav Nebeský
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Hamiltonian Connected ◽  
Connected Subgraphs

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

scholarly journals Large Induced Subgraphs with All Degrees Odd

1992 ◽  
Vol 1 (4) ◽  
pp. 335-349 ◽  
Author(s):  
A. D. Scott
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Connected Graph ◽  
Analogous Problem ◽  
Induced Subgraphs ◽  
Induced Subgraph

We prove that every connected graph of order n ≥ 2 has an induced subgraph with all degrees odd of order at least cn/log n, where cis a constant. We also give a bound in terms of chromatic number, and resolve the analogous problem for random graphs.

scholarly journals Removable Cycles Avoiding Two Connected Subgraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Y. M. Borse ◽  
B. N. Waphare
Keyword(s):  
Connected Graph ◽  
Sufficient Condition ◽  
Edge Disjoint ◽  
Connected Subgraphs

We provide a sufficient condition for the existence of a cycle in a connected graph which is edge-disjoint from two connected subgraphs and of such that is connected.

CO-MAXIMAL IDEAL GRAPHS OF COMMUTATIVE RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250114 ◽  
Author(s):  
MENG YE ◽  
TONGSUO WU
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Maximal Ideal ◽  
Connected Graph ◽  
Jacobson Radical ◽  
Clique Number ◽  
Commutative Rings ◽  
Maximal Ideals ◽  
Simple Connected Graph

In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use [Formula: see text] to denote this graph, with its vertices the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. We show some properties of this graph. For example, this graph is a simple, connected graph with diameter less than or equal to three, and both the clique number and the chromatic number of the graph are equal to the number of maximal ideals of the ring R.

THE RAMSEY NUMBER FOR A LINEAR FOREST VERSUS TWO IDENTICAL COPIES OF COMPLETE GRAPHS

2010 ◽  
Vol 02 (04) ◽  
pp. 437-444 ◽  
Author(s):  
I. WAYAN SUDARSANA ◽  
HILDA ASSIYATUN ◽  
ADIWIJAYA ◽  
SELVY MUSDALIFAH
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Ramsey Number ◽  
Complete Graphs ◽  
Cocktail Party ◽  
Linear Forest ◽  
Good For

Let H be a graph with the chromatic number h and the chromatic surplus s. A connected graph G of order n is called H-good if R(G, H) = (n - 1)(h - 1) + s. In this paper, we show that Pn is 2Km-good for n ≥ 3. Furthermore, we obtain the Ramsey number R(L, 2Km), where L is a linear forest. Moreover, we also give the Ramsey number R(L, Hm) which is an extension for R(kPn, Hm) proposed by Ali et al. [1], where Hm is a cocktail party graph on 2m vertices.

EQUITABLE COLORING OF 2-DEGENERATE GRAPH AND PLANE GRAPHS WITHOUT CYCLES OF SPECIFIC LENGTHS

2010 ◽  
Vol 02 (02) ◽  
pp. 207-211 ◽  
Author(s):  
YUEHUA BU ◽  
QIONG LI ◽  
SHUIMING ZHANG
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Connected Graph ◽  
Plane Graphs ◽  
Equitable Coloring ◽  
Odd Cycle ◽  
Equitable Chromatic Number

The equitable chromatic number χe(G) of a graph G is the smallest integer k for which G has a proper k-coloring such that the number of vertices in any two color classes differ by at most one. In 1973, Meyer conjectured that the equitable chromatic number of a connected graph G, which is neither a complete graph nor an odd cycle, is at most Δ(G). We prove that this conjecture holds for 2-degenerate graphs with Δ(G) ≥ 5 and plane graphs without 3, 4 and 5 cycles.

scholarly journals On resolving edge colorings in graphs

2003 ◽  
Vol 2003 (46) ◽  
pp. 2947-2959
Author(s):  
Varaporn Saenpholphat ◽  
Ping Zhang
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Edge Colorings

We study the relationships between the resolving edge chromatic number and other graphical parameters and provide bounds for the resolving edge chromatic number of a connected graph.

scholarly journals On local antimagic vertex coloring of corona products related to friendship and fan graph

2021 ◽  
Vol 5 (2) ◽  
pp. 110
Author(s):  
Zein Rasyid Himami ◽  
Denny Riama Silaban
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Vertex Coloring ◽  
Antimagic Labeling ◽  
Minimum Number ◽  
Fan Graph ◽  
Corona Product

Let <em>G</em>=(<em>V</em>,<em>E</em>) be connected graph. A bijection <em>f </em>: <em>E</em> → {1,2,3,..., |<em>E</em>|} is a local antimagic of <em>G</em> if any adjacent vertices <em>u,v</em> ∈ <em>V</em> satisfies <em>w</em>(<em>u</em>)≠ <em>w</em>(<em>v</em>), where <em>w</em>(<em>u</em>)=∑<sub>e∈E(u) </sub><em>f</em>(<em>e</em>), <em>E</em>(<em>u</em>) is the set of edges incident to <em>u</em>. When vertex <em>u</em> is assigned the color <em>w</em>(<em>u</em>), we called it a local antimagic vertex coloring of <em>G</em>. A local antimagic chromatic number of <em>G</em>, denoted by <em>χ</em><sub>la</sub>(<em>G</em>), is the minimum number of colors taken over all colorings induced by the local antimagic labeling of <em>G</em>. In this paper, we determine the local antimagic chromatic number of corona product of friendship and fan with null graph on <em>m</em> vertices, namely, <em>χ</em><sub>la</sub>(<em>F</em><sub>n</sub> ⊙ \overline{K_m}) and <em>χ</em><sub>la</sub>(<em>f</em><sub>(1,n)</sub> ⊙ \overline{K_m}).

scholarly journals Strengthened Brooks' Theorem for Digraphs of Girth at least Three

10.37236/682 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Ararat Harutyunyan ◽  
Bojan Mohar
Keyword(s):  
Chromatic Number ◽  
Complete Graph ◽  
Connected Graph ◽  
Absolute Constant ◽  
Maximum Degree ◽  
Geometric Mean ◽  
Degree Of A Vertex ◽  
Odd Cycle ◽  
Directed Cycles

Brooks' Theorem states that a connected graph $G$ of maximum degree $\Delta$ has chromatic number at most $\Delta$, unless $G$ is an odd cycle or a complete graph. A result of Johansson shows that if $G$ is triangle-free, then the chromatic number drops to $O(\Delta / \log \Delta)$. In this paper, we derive a weak analog for the chromatic number of digraphs. We show that every (loopless) digraph $D$ without directed cycles of length two has chromatic number $\chi(D) \leq (1-e^{-13}) \tilde{\Delta}$, where $\tilde{\Delta}$ is the maximum geometric mean of the out-degree and in-degree of a vertex in $D$, when $\tilde{\Delta}$ is sufficiently large. As a corollary it is proved that there exists an absolute constant $\alpha < 1$ such that $\chi(D) \leq \alpha (\tilde{\Delta} + 1)$ for every $\tilde{\Delta} > 2$.

scholarly journals A Similarity Measurement Method Based on Graph Kernel for Disconnected Graphs

2019 ◽  
Author(s):  
Jun Gao ◽  
Jianliang Gao
Keyword(s):  
Real World ◽  
Connected Graph ◽  
Measurement Method ◽  
Dimensional Space ◽  
Similarity Measurement ◽  
Graph Kernel ◽  
The Real ◽  
Connected Subgraphs ◽  
Low Dimensional ◽  
Disconnected Graphs

Disconnected graphs are very common in the real world. However, most existing methods for graph similarity focus on connected graph. In this paper, we propose an effective approach for measuring the similarity of disconnected graphs. By embedding connected subgraphs with graph kernel, we obtain the feature vectors in low dimensional space. Then, we match the subgraphs and weigh the similarity of matched subgraphs. Finally, an intuitive example shows the feasibility of the method.

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