Density of universal classes of series-parallel graphs

International audience A class of graphs $\mathcal{C}$ ordered by the homomorphism relation is universal if every countable partial order can be embedded in $\mathcal{C}$. It was shown in [ZH] that the class $\mathcal{C_k}$ of $k$-colorable graphs, for any fixed $k≥3$, induces a universal partial order. In [HN1], a surprisingly small subclass of $\mathcal{C_3}$ which is a proper subclass of $K_4$-minor-free graphs $(\mathcal{G/K_4)}$ is shown to be universal. In another direction, a density result was given in [PZ], that for each rational number $a/b ∈[2,8/3]∪ \{3\}$, there is a $K_4$-minor-free graph with circular chromatic number equal to $a/b$. In this note we show for each rational number $a/b$ within this interval the class $\mathcal{K_{a/b}}$ of $0K_4$-minor-free graphs with circular chromatic number $a/b$ is universal if and only if $a/b ≠2$, $5/2$ or $3$. This shows yet another surprising richness of the $K_4$-minor-free class that it contains universal classes as dense as the rational numbers.

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Construction of Kn-minor free graphs with given circular chromatic number

Discrete Mathematics ◽

10.1016/s0012-365x(02)00529-0 ◽

2003 ◽

Vol 263 (1-3) ◽

pp. 191-206 ◽

Cited By ~ 7

Author(s):

Sheng-Chyang Liaw ◽

Zhishi Pan ◽

Xuding Zhu

Keyword(s):

Chromatic Number ◽

Circular Chromatic Number ◽

Minor Free Graphs

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Circular colouring and graph homomorphism

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032627 ◽

1999 ◽

Vol 59 (1) ◽

pp. 83-97 ◽

Cited By ~ 9

Author(s):

Xuding zhu

Keyword(s):

Chromatic Number ◽

Rational Number ◽

Sharp Contrast ◽

Complete Graphs ◽

Graph Homomorphism ◽

Circular Chromatic Number

For any pair of integers p, q such that (p, q) = 1 and p ≥ 2q, the graph has vertices {0, 1, …, p − 1} and edges {ij: q ≤ |i − j| ≤ p − q}. These graphs play the same role in the study of circular chromatic number as that played by the complete graphs in the study of chromatic number. The graphs share many properties of the complete graphs. However, there are also striking differences between the graphs and the complete graphs. We shall prove in this paper that for many pairs of integers p, q, one may delete most of the edges of so that the resulting graph still has circular chromatic number p/q. To be precise, we shall prove that for any number r < 2, there exists a rational number p/q (where (p, q) = 1) which is less than r but arbitrarily close to r, such that contains a subgraph H with and . This is in sharp contrast to the fact that the complete graphs are edge critical, that is, the deletion of any edge will decrease its chromatic number and its circular chromatic number.

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Graph Powers and Graph Homomorphisms

The Electronic Journal of Combinatorics ◽

10.37236/289 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 10

Author(s):

Hossein Hajiabolhassan ◽

Ali Taherkhani

Keyword(s):

Bipartite Graph ◽

Chromatic Number ◽

Rational Number ◽

Complete Graph ◽

Rational Numbers ◽

Sufficient Condition ◽

Fractional Powers ◽

Circular Chromatic Number ◽

Equivalent Definition ◽

Graph Homomorphisms

In this paper, we investigate some basic properties of fractional powers. In this regard, we show that for any non-bipartite graph $G$ and positive rational numbers ${2r+1\over 2s+1} < {2p+1\over 2q+1}$, we have $G^{2r+1\over 2s+1} < G^{2p+1\over 2q+1}$. Next, we study the power thickness of $G$, that is, the supremum of rational numbers ${2r+1\over 2s+1}$ such that $G$ and $G^{2r+1\over 2s+1}$ have the same chromatic number. We prove that the power thickness of any non-complete circular complete graph is greater than one. This provides a sufficient condition for the equality of the chromatic number and the circular chromatic number of graphs. Finally, we introduce an equivalent definition for the circular chromatic number of graphs in terms of fractional powers. Also, we show that for any non-bipartite graph $G$ if $0 < {{2r+1}\over {2s+1}} \leq {{\chi(G)}\over{3(\chi(G)-2)}}$, then $\chi(G^{{2r+1}\over {2s+1}})=3$. Moreover, $\chi(G)\neq\chi_c(G)$ if and only if there exists a rational number ${{2r+1}\over {2s+1}}>{{\chi(G)}\over{3(\chi(G)-2)}}$ for which $\chi(G^{{2r+1}\over {2s+1}})= 3$.

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Another Infinite Sequence of Dense Triangle-Free Graphs

The Electronic Journal of Combinatorics ◽

10.37236/1381 ◽

1998 ◽

Vol 5 (1) ◽

Cited By ~ 2

Author(s):

Stephan Brandt ◽

Tomaž Pisanski

Keyword(s):

Structural Properties ◽

Chromatic Number ◽

Infinite Sequence ◽

Minimum Degree ◽

Free Graph ◽

The Core ◽

Graph Homomorphisms ◽

Natural Way

The core is the unique homorphically minimal subgraph of a graph. A triangle-free graph with minimum degree $\delta > n/3$ is called dense. It was observed by many authors that dense triangle-free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs in this sequence are 3-colourable. Only two additional cores with chromatic number 4 were known. We show that the additional graphs are the initial terms of a second infinite sequence of cores.

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The b-chromatic number of power graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.333 ◽

2003 ◽

Vol Vol. 6 no. 1 ◽

Author(s):

Brice Effantin ◽

Hamamache Kheddouci

Keyword(s):

Chromatic Number ◽

Binary Trees ◽

Proper Coloring ◽

Power Cycles ◽

International Audience

International audience The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles.

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Cubical coloring — fractional covering by cuts and semidefinite programming

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2134 ◽

2015 ◽

Vol Vol. 17 no.2 (Graph Theory) ◽

Author(s):

Robert Šámal

Keyword(s):

Semidefinite Programming ◽

Chromatic Number ◽

Fractional Chromatic Number ◽

Maximum Cut ◽

Continuous Mappings ◽

Polynomial Time Approximation Algorithm ◽

Eigenvalue Bound ◽

International Audience ◽

Graph Parameters ◽

Fractional Covering

International audience We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

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