On the restricted homomorphism problem

Brewster, Richard C.; Graves, Timothy

doi:10.1016/j.dam.2007.03.028

Cited by 2 publications

(2 citation statements)

References 13 publications

(11 reference statements)

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“…The H ‐homomorphism graph is perhaps better known as the reflexive subgraph of the exponential graph

H^{G}

. Exponential graphs were introduced by Lovász in and have been used to study and solve many interesting homomorphism and coloring problems, see for example [, ]. For loop‐free graphs, the connected components of

{scriptC}_{H} (G)

correspond to mixing classes , whereas the connected components of

{scriptH}_{H} (G)

correspond to homotopy classes .…”

Section: Homotopymentioning

confidence: 99%

Mixing Homomorphisms, Recolorings, and Extending Circular Precolorings

Brewster

Noel

2014

Journal of Graph Theory

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This work brings together ideas of mixing graph colorings, discrete homotopy, and precoloring extension. A particular focus is circular colorings. We prove that all the (k,q)‐colorings of a graph G can be obtained by successively recoloring a single vertex provided k/q≥2col(G) along the lines of Cereceda, van den Heuvel, and Johnson's result for k‐colorings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colorings coincide. As a corollary, we obtain an Albertson‐type extension theorem for (k,q)‐precolorings of circular cliques. Such a result was first conjectured by Albertson and West. General results on homomorphism mixing are presented, including a characterization of graphs G for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.

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“…The H ‐homomorphism graph is perhaps better known as the reflexive subgraph of the exponential graph

H^{G}

{scriptC}_{H} (G)

correspond to mixing classes , whereas the connected components of

{scriptH}_{H} (G)

correspond to homotopy classes .…”

Section: Homotopymentioning

confidence: 99%

Mixing Homomorphisms, Recolorings, and Extending Circular Precolorings

Brewster

Noel

2014

Journal of Graph Theory

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show abstract

“…The Hhomomorphism graph is perhaps better known as the reflexive subgraph of the exponential graph H G . Exponential graphs were introduced by Lovász in [22] and have been used to study and solve many interesting homomorphism and colouring problems, see for example [9,17]. For loop-free graphs, the connected components of C H (G) correspond to mixing classes, whereas the connected components of H H (G) correspond to homotopy classes.…”

Section: Clearly the Restriction Of H To Gmentioning

confidence: 99%

Mixing Homomorphisms, Recolourings, and Extending Circular Precolourings

Brewster,

Noel

2014

Preprint

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This work brings together ideas of mixing graph colourings, discrete homotopy, and precolouring extension. A particular focus is circular colourings. We prove that all the (k, q)-colourings of a graph G can be obtained by successively recolouring a single vertex provided k/q ≥ 2col(G) along the lines of Cereceda, van den Heuvel and Johnson's result for kcolourings. We give various bounds for such mixing results and discuss their sharpness, including cases where the bounds for circular and classical colourings coincide. As a corollary, we obtain an Albertson-type extension theorem for (k, q)-precolourings of circular cliques. Such a result was first conjectured by Albertson and West.General results on homomorphism mixing are presented, including a characterization of graphs G for which the endomorphism monoid can be generated through the mixing process. As in similar work of Brightwell and Winkler, the concept of dismantlability plays a key role.

show abstract

On the restricted homomorphism problem

Cited by 2 publications

References 13 publications

Mixing Homomorphisms, Recolorings, and Extending Circular Precolorings

Mixing Homomorphisms, Recolorings, and Extending Circular Precolorings

Mixing Homomorphisms, Recolourings, and Extending Circular Precolourings

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