A bound on the chromatic number using the longest odd cycle length

2007 ◽  
Vol 54 (4) ◽  
pp. 267-276 ◽  
Author(s):  
Sreyash Kenkre ◽  
Sundar Vishwanathan
Keyword(s):  
Chromatic Number ◽  
Cycle Length ◽  
Odd Cycle

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 Cycle length parities and the chromatic number

2009 ◽  
pp. n/a-n/a
Author(s):  
Christian Löwenstein ◽  
Dieter Rautenbach ◽  
Ingo Schiermeyer
Keyword(s):  
Chromatic Number ◽  
Cycle Length

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 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 Note on Chromatic Number and Induced Odd Cycles

10.37236/5555 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Baogang Xu ◽  
Gexin Yu ◽  
Xiaoya Zha
Keyword(s):  
Chromatic Number ◽  
Bipartite Subgraph ◽  
Odd Cycle ◽  
Odd Cycles

An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyárfás and proved that if a graph $G$ has no odd holes then $\chi(G)\le 2^{2^{\omega(G)+2}}$. Chudnovsky, Robertson, Seymour and Thomas showed that if $G$ has neither $K_4$ nor odd holes then $\chi(G)\le 4$. In this note, we show that if a graph $G$ has neither triangles nor quadrilaterals, and has no odd holes of length at least 7, then $\chi(G)\le 4$ and $\chi(G)\le 3$ if $G$ has radius at most $3$, and for each vertex $u$ of $G$, the set of vertices of the same distance to $u$ induces a bipartite subgraph. This answers some questions in Plummer and Zha (2014).

scholarly journals On the Alon-Tarsi Number and Chromatic-Choosability of Cartesian Products of Graphs

10.37236/7740 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Hemanshu Kaul ◽  
Jeffrey A. Mudrock
Keyword(s):  
Chromatic Number ◽  
Cartesian Product ◽  
Hamiltonian Path ◽  
Complete Graphs ◽  
Cartesian Products ◽  
Hamilton Path ◽  
Odd Cycle ◽  
Delta G ◽  
List Chromatic Number ◽  
Cartesian Products Of Graphs

We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest $k$ for which there is an orientation, $D$, of $G$ with max indegree $k\!-\!1$ such that the number of even and odd circulations contained in $D$ are different. It is known that $\chi(G) \leq \chi_\ell(G) \leq \chi_p(G) \leq AT(G)$, where  $\chi(G)$ is the chromatic number, $\chi_\ell(G)$ is the list chromatic number, and $\chi_p(G)$ is the paint number of $G$. In this paper we find families of graphs $G$ and $H$ such that $\chi(G \square H) = AT(G \square H)$, reducing this sequence of inequalities to equality. We show that the Alon-Tarsi number of the Cartesian product of an odd cycle and a path is always equal to 3. This result is then extended to show that if $G$ is an odd cycle or a complete graph and $H$ is a graph on at least two vertices containing the Hamilton path $w_1, w_2, \ldots, w_n$ such that for each $i$, $w_i$ has a most $k$ neighbors among $w_1, w_2, \ldots, w_{i-1}$, then $AT(G \square H) \leq \Delta(G)+k$ where $\Delta(G)$ is the maximum degree of $G$.  We discuss other extensions for $G \square H$, where $G$ is such that $V(G)$ can be partitioned into odd cycles and complete graphs, and $H$ is a graph containing a Hamiltonian path. We apply these bounds to get chromatic-choosable Cartesian products, in fact we show that these families of graphs have $\chi(G) = AT(G)$, improving previously known bounds.

scholarly journals Distance Colouring Without One Cycle Length

2018 ◽  
Vol 27 (5) ◽  
pp. 794-807 ◽  
Author(s):  
ROSS J. KANG ◽  
FRANÇOIS PIROT
Keyword(s):  
Maximum Degree ◽  
Cycle Length ◽  
Open Problems ◽  
Logarithmic Factor ◽  
Odd Cycle ◽  
Vertex Colouring ◽  
Edge Colouring

We consider distance colourings in graphs of maximum degree at most d and how excluding one fixed cycle of length ℓ affects the number of colours required as d → ∞. For vertex-colouring and t ⩾ 1, if any two distinct vertices connected by a path of at most t edges are required to be coloured differently, then a reduction by a logarithmic (in d) factor against the trivial bound O(dt) can be obtained by excluding an odd cycle length ℓ ⩾ 3t if t is odd or by excluding an even cycle length ℓ ⩾ 2t + 2. For edge-colouring and t ⩾ 2, if any two distinct edges connected by a path of fewer than t edges are required to be coloured differently, then excluding an even cycle length ℓ ⩾ 2t is sufficient for a logarithmic factor reduction. For t ⩾ 2, neither of the above statements are possible for other parity combinations of ℓ and t. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).

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.

On the sigma chromatic number of the join of a finite number of paths and cycles

2019 ◽  
pp. 2150019
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Finite Number ◽  
Chromatic Number ◽  
Connected Graph ◽  
Odd Cycle ◽  
Minimum Number ◽  
Simple Connected Graph

Let [Formula: see text] be a simple connected graph and [Formula: see text] a coloring of the vertices in [Formula: see text] For any [Formula: see text], let [Formula: see text] be the sum of colors of the vertices adjacent to [Formula: see text]. Then [Formula: see text] is called a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] The minimum number of colors needed in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text], denoted by [Formula: see text] In this paper, we prescribe a sigma coloring of the join of paths and cycles. As a consequence, we determine the sigma chromatic number of the join of a finite number of paths and cycles. In particular, let [Formula: see text], where [Formula: see text] or [Formula: see text] with [Formula: see text] If [Formula: see text], where [Formula: see text] and [Formula: see text], then [Formula: see text] if [Formula: see text] is an odd cycle, for some [Formula: see text] and [Formula: see text] otherwise.

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