Homomorphism-homogeneous L-colored graphs

arXiv (Cornell University), 2014

We address the problem of characterizing H-coloring problems that are first-order definable on a fixed class of relational structures. In this context, we give also several characterizations of a homomorphism dualities arising in a class of structure.

Journal of Combinatorial Theory, 2000

The homomorphisms of oriented or undirected graphs, the oriented chromatic number, the relationship between acyclic coloring number and oriented chromatic number, have been recently studied in 1, 3, 5, 6, 8, 10, 11]. Improving and combining earlier techniques of Alon and Marshall 1] and Raspaud and Sopena 10], we prove here a general result about homomorphisms of colored mixed graphs which implies all these earlier results about planar graphs. We also determine the exact chromatic number of colored mixed trees. For this, we introduce the notion of colored homomorphism for mixed graphs containing both colored arcs and colored edges. 2

Discrete Mathematics, 2002

For every pair of ÿnite connected graphs F and H , and every positive integer k, we construct a universal graph U with the following properties: (1) There is a homomorphism : U → H , but no homomorphism from F to U. (2) For every graph G with maximum degree no more than k having a homomorphism h : G → H , but no homomorphism from F to G, there is a homomorphism : G → U , such that h = •. Particularly, this solves a problem presented in [1] and [2] regarding the chromatic number of a universal graph.

European Journal of Combinatorics

In this paper we study the existence of homomorphisms G → H using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number t ≥ 2 for which there exists an assignment of unit vectors i → pi to its vertices such that pi, pj ≤ −1/(t − 1), when i ∼ j. Our approach allows to reprove, without using the Erdős-Ko-Rado Theorem, that for n > 2r the Kneser graph Kn:r and the q-Kneser graph qKn:r are cores, and furthermore, that for n/r = n ′ /r ′ there exists a homomorphism Kn:r → K n ′ :r ′ if and only if n divides n ′ . In terms of new applications, we show that the even-weight component of the distance k-graph of the n-cube H n,k is a core and also, that non-bipartite Taylor graphs are cores. Additionally, we give a necessary and sufficient condition for the existence of homomorphisms H n,k → H n ′ ,k ′ when n/k = n ′ /k ′ . Lastly, we show that if a 2-walk-regular graph (which is non-bipartite and not complete multipartite) has a unique optimal vector coloring, it is a core. Based on this sufficient condition we conducted a computational study on Ted Spence's list of strongly regular graphs and found that at least 84% are cores.

Graphs and Combinatorics, 2019

Indicated coloring is a type of game coloring in which two players collectively color the vertices of a graph in the following way. In each round the first player (Ann) selects a vertex, and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben's strategy) is called the indicated chromatic number of G, denoted by χ i (G). In this paper, we obtain structural characterization of connected {P 5 , K 4 , Kite, Bull}-free graphs which contains an induced C 5 and connected {P 6 , C 5 , K 1,3 }-free graphs that contains an induced C 6 . Also, we prove that {P 5 , K 4 , Kite, Bull}-free graphs that contains an induced C 5 and {P 6 , C 5 , P 5 , K 1,3 }free graphs which contains an induced C 6 are k-indicated colorable for all k ≥ χ(G). In addition, we show that K[C 5 ] is k-indicated colorable for all k ≥ χ(G) and as a consequence, we exhibit that {P 2 ∪ P 3 , C 4 }-free graphs, {P 5 , C 4 }-free graphs are k-indicated colorable for all k ≥ χ(G). This partially answers one of the questions which was raised by A. Grzesik in .

Discrete Mathematics, 2008

A graph is point determining if distinct vertices have distinct neighbourhoods. A realization of a point determining graph H is a point determining graph G such that each vertex-removed subgraph G − x which is point determining, is isomorphic to H. We study the fine structure of point determining graphs, and conclude that every point determining graph has at most two realizations. A full homomorphism of a graph G to a graph H is a vertex mapping f such that for distinct vertices u and v of G, we have uv an edge of G if and only if f (u)f (v) is an edge of H. For a fixed graph H, a full H-colouring of G is a full homomorphism of G to H. A minimal H-obstruction is a graph G which does not admit a full H-colouring, such that each proper induced subgraph of G admits a full H-colouring. We analyse minimal H-obstructions using our results on point determining graphs. We connect the two problems by proving that if H has k vertices, then a graph with k + 1 vertices is a minimal H-obstruction if and only if it is a realization of H. We conclude that every minimal H-obstruction has at most k + 1 vertices, and there are at most two minimal H-obstructions with k + 1 vertices. We also consider full homomorphisms to graphs H in which loops are allowed. If H has loops and k vertices without loops, then every minimal H-obstruction has at most (k + 1)(+ 1) vertices, and, when both k and are positive, there is at most one minimal H-obstruction with (k + 1)(+ 1) vertices. In particular, this yields a finite forbidden subgraph characterization of full H-colourability, for any graph H with loops allowed.

2019

We present a result showing that any countably infinite HH-homogeneous graph that does not contain the Rado graph as a spanning subgraph has finite independence number; from this we derive a classification of MB-homogeneous graphs. Additionally, we present constructions that yield new HH-homogeneous graphs. Homomorphism-homogeneity was introduced in [3] as a variation on ultrahomogeneity. A relational structure G is homomorphism-homogeneous if every homomorphism f between finite induced substructures is the restriction of an endomorphism F of G to the domain of f . This definition can be refined by specifying what kind of homomorphism is f and what kind of endomorphism F is; Lockett and Truss introduced many of these classes in [6]. Following the tradition, we specify a morphism-extension class by two characters XY, where X comes from {H,M, I} (the letters stand for Homomorphism, Monomorphism, and Isomorphism) and Y comes from {H, I,A,B,M} (corresponding to endomorphism, Isomorphism...

Journal of Combinatorial Theory, Series B, 2004

We prove that for every graph H and positive integers k and l there exists a graph G with girth at least l such that for all graphs H 0 with at most k vertices there exists a homomorphism G-H 0 if and only if there exists a homomorphism H-H 0 : This implies (for H ¼ K k) the classical result of Erd + os and other generalizations (such as Sparse Incomparability Lemma). We refine the above statement to the 1-1 correspondence between the set of all homomorphisms G-H 0 and the set of all homomorphisms H-H 0 : This in turn yields the existence of sparse uniquely H-colorable graphs and, perhaps surprisingly, provides a characterization of the graphs H for which the analog of Mu¨ller's theorem holds for H-colorings.

NATO ASI Series C Mathematical and …, 1997

This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We discuss vertex transitive graphs and Cayley graphs and their rather fundamental role in some aspects of graph homomorphisms. Graph colourings are then explored as homomorphisms, followed by a discussion of various graph products.

Annals of the New York Academy of Sciences, 1979

Quaestiones Mathematicae, 2017

A graph property is any isomorphism-closed class of graphs. A property P is hereditary if, whenever a graph G is in P, and H is a subgraph of G, then H is also in P. For a hereditary graph property P, positive integer l and a graph G, let ρ ′ P,l (G) be the minimum number of colours needed to colour the edges of G, such that any subgraph of G induced by edges coloured with (at most) l colours is in P. We study the properties S k and E k , where S k contains all graphs of maximum degree at most k and E k contains graphs, whose components have at most k edges. We present results on ρ ′ S k ,l (G) and ρ ′ E k ,l (G) for various graphs G. We prove that for graphs G of maximum degree ∆ we have ⌈

Combinatorica, 2007

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation m•n ≡ m+n-2, the lengths of omitted cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We conjecture that all such monoids are realized, and prove that several are. We then characterize exactly Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having 2, 4, and 5 elements, respectively. We then characterize, in two quite different ways, the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. And the second characterization is by means of a homomorphism duality Contents

2016

The Compactness Theorem for graph colourings, by De Bruijn and Erdős, can be restated as follows. If all the finite subgraphs of a graph G are homomorphic to a finite complete graph Kn, then G is also homomorphic to Kn. In short, finite complete graphs have the following interesting quality: a graph G is not homomorphic to a complete graph if and only if some finite subgraph of G is not homomorphic to said complete graph. There have been many investigations into graphs H that posses this remarkable characteristic of complete graphs. We further this investigation and describe a graph with finite chromatic number that does not posses the aforementioned quality. Our approach is from a lattice theoretic stand point. That is to say we will study those sets of graphs that are homomorphic to a specific graph. Such sets we call hom-properties, and when a graph possesses

arXiv: Combinatorics, 2013

A \emph{directional labeling} of an edge $\emph{uv}$ in a graph $G=(V,E)$ by an ordered pair $ab$ is a labeling of the edge $uv$ such that the label on $uv$ in the direction from $u$ to $v$ is $\ell(uv)=ab$, and $\ell(vu)=ba$. New characterizations of 2-colorable (bipartite) and 3-colorable graphs are obtained in terms of directional labeling of edges of a graph by ordered pairs $ab$ and $ba$. In addition we obtain characterizations of 2-colorable and 3-colorable graphs in terms of matrices called directional adjacency matrices.

Combinatorics, Probability and Computing, 2006

We study relational structures (especially graphs and posets) which satisfy the analogue of homogeneity but for homomorphisms rather than isomorphisms. The picture is rather different. Our main results are partial characterisations of countable graphs and posets with this property; an analogue of Fraïssé's Theorem; and representations of monoids as endomorphism monoids of such structures.