10,949 research outputs found
The chromatic index of strongly regular graphs
We determine (partly by computer search) the chromatic index (edge-chromatic
number) of many strongly regular graphs (SRGs), including the SRGs of degree and their complements, the Latin square graphs and their complements,
and the triangular graphs and their complements. Moreover, using a recent
result of Ferber and Jain it is shown that an SRG of even order , which is
not the block graph of a Steiner 2-design or its complement, has chromatic
index , when is big enough. Except for the Petersen graph, all
investigated connected SRGs of even order have chromatic index equal to their
degree, i.e., they are class 1, and we conjecture that this is the case for all
connected SRGs of even order.Comment: 10 page
Acyclic edge coloring of subcubic graphs
AbstractAn acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors
On the equitable total (k+1)-coloring of k-regular graphs
A graph is considered to be totally colored when one color is assigned to each vertex and to each edge so that no adjacent or incident vertices or edges bear the same color. The \textit{total chromatic number} of a graph is the least number of colors required to totally color a graph. This paper focuses on -regular graphs, whose symmetry and regularity allow for a closer look at general total coloring strategies. Such graphs include the previously defined M\ obius ladder, which has a total chromatic number of 5, as well as the newly defined bird\u27s nest, which is shown to have a total chromatic number of 4. Furthermore, a total 4-coloring of the Petersen graph is examined and the total -colorings of -regular graphs is discussed. More specifically, it is proposed that any -coloring of any -regular graph is inherently equitable for all given a bound on the order of the graph. That is to say that every color is used no more than one time more than any other color when totally coloring the graph
The b-Chromatic Number of Regular Graphs via The Edge Connectivity
\noindent The b-chromatic number of a graph , denoted by , is the
largest integer that admits a proper coloring by colors, such that
each color class has a vertex that is adjacent to at least one vertex in each
of the other color classes. El Sahili and Kouider [About b-colorings of regular
graphs, Res. Rep. 1432, LRI, Univ. Orsay, France, 2006] asked whether it is
true that every -regular graph of girth at least 5 satisfies
. Blidia, Maffray, and Zemir [On b-colorings in regular graphs,
Discrete Appl. Math. 157 (2009), 1787-1793] showed that the Petersen graph
provides a negative answer to this question, and then conjectured that the
Petersen graph is the only exception. In this paper, we investigate a
strengthened form of the question.
The edge connectivity of a graph , denoted by , is the minimum
cardinality of a subset of such that is either
disconnected or a graph with only one vertex. A -regular graph is called
super-edge-connected if every minimum edge-cut is the set of all edges incident
with a vertex in , i.e., and every minimum edge-cut of
isolates a vertex. We show that if is a -regular graph that contains no
4-cycle, then whenever is not super-edge-connected
Extremal problems involving forbidden subgraphs
In this thesis, we study extremal problems involving forbidden subgraphs. We are interested in extremal problems over a family of graphs or over a family of hypergraphs.
In Chapter 2, we consider improper coloring of graphs without short cycles. We find how sparse an improperly critical graph can be when it has no short cycle. In particular, we find the exact threshold of density of triangle-free -colorable graphs and we find the asymptotic threshold of density of -colorable graphs of large girth when .
In Chapter 3, we consider other variations of graph coloring. We determine harmonious chromatic number of trees with large maximum degree and show upper bounds of -dynamic chromatic number of graphs in terms of other parameters.
In Chapter 4, we consider how dense a hypergraph can be when we forbid some subgraphs.
In particular, we characterize hypergraphs with the maximum number of edges that contain no -regular subgraphs. We also establish upper bounds for the number of edges in graphs and hypergraphs with no edge-disjoint equicovering subgraphs
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