scholarly journals Beyond Degree Choosability

10.37236/6179 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Maximum Degree ◽  
List Coloring ◽  
Odd Cycle ◽  
Eulerian Subgraphs

Let $G$ be a connected graph with maximum degree $\Delta$. Brooks' theorem states that $G$ has a $\Delta$-coloring unless $G$ is a complete graph or an odd cycle. A graph $G$ is degree-choosable if $G$ can be properly colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph $G$ is degree-choosable unless each block of $G$ is a complete graph or an odd cycle; such a graph $G$ is a Gallai tree. This degree-choosability result was further strengthened to Alon—Tarsi orientations; these are orientations of $G$ in which the number of spanning Eulerian subgraphs with an even number of edges differs from the number with an odd number of edges. A graph $G$ is degree-AT if $G$ has an Alon—Tarsi orientation in which each vertex has indegree at least 1. Alon and Tarsi showed that if $G$ is degree-AT, then $G$ is also degree-choosable. Hladký, Král', and Schauz showed that a connected graph is degree-AT if and only if it is not a Gallai tree. In this paper, we consider pairs $(G,x)$ where $G$ is a connected graph and $x$ is some specified vertex in $V(G)$. We characterize pairs such that $G$ has no Alon—Tarsi orientation in which each vertex has indegree at least 1 and $x$ has indegree at least 2. When $G$ is 2-connected, the characterization is simple to state.

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 $\tilde{\Delta} > 2$.

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.

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 Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Gábor Bacsó ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Connected Graph ◽  
Maximum Degree ◽  
Free Graph ◽  
Polynomial Time Algorithms ◽  
Odd Cycle ◽  
Vertex Set ◽  
International Audience

Graphs and Algorithms International audience A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

Strongly Projective Graphs

2002 ◽  
Vol 54 (4) ◽  
pp. 757-768
Author(s):  
Benoit Larose
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Complete Graphs ◽  
Projective Graph ◽  
Odd Cycle ◽  
Odd Cycles

AbstractWe introduce the notion of strongly projective graph, and characterise these graphs in terms of their neighbourhood poset. We describe certain exponential graphs associated to complete graphs and odd cycles. We extend and generalise a result of Greenwell and Lovász [6]: if a connected graph G does not admit a homomorphism to K, where K is an odd cycle or a complete graph on at least 3 vertices, then the graph G × Ks admits, up to automorphisms of K, exactly s homomorphisms to K.

scholarly journals Strengthening $(a,b)$-Choosability Results to $(a,b)$-Paintability

10.37236/7163 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Thomas Mahoney
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Online Version ◽  
Independent Subset ◽  
Odd Cycle ◽  
Disjoint Sets ◽  
Delta G ◽  
To Receive

Let $a,b\in\mathbb{N}$. A graph $G$ is $(a,b)$-choosable if for any list assignment $L$ such that $|L(v)|\ge a$, there exists a coloring in which each vertex $v$ receives a set $C(v)$ of $b$ colors such that $C(v)\subseteq L(v)$ and $C(u)\cap C(w)=\emptyset$ for any $uw\in E(G)$. In the online version of this problem, on each round, a set of vertices allowed to receive a particular color is marked, and the coloring algorithm chooses an independent subset of these vertices to receive that color. We say $G$ is $(a,b)$-paintable if when each vertex $v$ is allowed to be marked $a$ times, there is an algorithm to produce a coloring in which each vertex $v$ receives $b$ colors such that adjacent vertices receive disjoint sets of colors.We show that every odd cycle $C_{2k+1}$ is $(a,b)$-paintable exactly when it is $(a,b)$-chosable, which is when $a\ge2b+\lceil b/k\rceil$. In 2009, Zhu conjectured that if $G$ is $(a,1)$-paintable, then $G$ is $(am,m)$-paintable for any $m\in\mathbb{N}$. The following results make partial progress towards this conjecture. Strengthening results of Tuza and Voigt, and of Schauz, we prove for any $m \in \mathbb{N}$ that $G$ is $(5m,m)$-paintable when $G$ is planar. Strengthening work of Tuza and Voigt, and of Hladky, Kral, and Schauz, we prove that for any connected graph $G$ other than an odd cycle or complete graph and any $m\in\mathbb{N}$, $G$ is $(\Delta(G)m,m)$-paintable.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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]

COMPUTING THE RUPTURE DEGREE IN COMPOSITE GRAPHS

2010 ◽  
Vol 21 (03) ◽  
pp. 311-319 ◽  
Author(s):  
AYSUN AYTAC ◽  
ZEYNEP NIHAN ODABAS
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
The Other ◽  
Trade Off ◽  
Number Of Components ◽  
Np Complete ◽  
Work Done ◽  
Better Than

The rupture degree of an incomplete connected graph G is defined by [Formula: see text] where w(G - S) is the number of components of G - S and m(G - S) is the order of a largest component of G - S. For the complete graph Kn, rupture degree is defined as 1 - n. This parameter can be used to measure the vulnerability of a graph. Rupture degree can reflect the vulnerability of graphs better than or independent of the other parameters. To some extent, it represents a trade-off between the amount of work done to damage the network and how badly the network is damaged. Computing the rupture degree of a graph is NP-complete. In this paper, we give formulas for the rupture degree of composition of some special graphs and we consider the relationships between the rupture degree and other vulnerability parameters.

scholarly journals On Decomposing Graphs of Large Minimum Degree into Locally Irregular Subgraphs

10.37236/5173 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Connected Graph ◽  
Special Class ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Probabilistic Approach ◽  
The Family ◽  
Edge Colourings ◽  
Odd Cycles

A locally irregular graph is a graph whose adjacent vertices have distinct degrees. We say that a graph G can be decomposed into k locally irregular subgraphs if its edge set may be partitioned into k subsets each of which induces a locally irregular subgraph in G. It has been conjectured that apart from the family of exceptions which admit no such decompositions, i.e., odd paths, odd cycles and a special class of graphs of maximum degree 3, every connected graph can be decomposed into 3 locally irregular subgraphs. Using a combination of a probabilistic approach and some known theorems on degree constrained subgraphs of a given graph, we prove this to hold for graphs of minimum degree at least $10^{10}$. This problem is strongly related to edge colourings distinguishing neighbours by the pallets of their incident colours and to the 1-2-3 Conjecture. In particular, the contribution of this paper constitutes a strengthening of a result of Addario-Berry, Aldred, Dalal and Reed [J. Combin. Theory Ser. B 94 (2005) 237-244].

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