Complex and homomorphic chromatic number of signed planar simple graphs - Archive ouverte HAL
Article Dans Une Revue Graphs and Combinatorics Année : 2022

Complex and homomorphic chromatic number of signed planar simple graphs

Résumé

We introduce the notion of complex chromatic number of signed graphs as follows: given the set C k,l = {±1, ±2,. .. , ±k} ∪ {±1i, ±2i,. .. , ±li}, where i = √ −1, a signed graph (G, σ) is said to be (k, l)-colorable if there exists a mapping c of vertices of G to C k,l such that for every edge xy of G we have c(x)c(y) = σ(xy)|c(x) 2 |. The complex chromatic number of a signed graph (G, σ), denoted χ com (G, σ), is defined to be the smallest order of C k,l such that (G, σ) admits a (k, l)-coloring. In this work, after providing an equivalent definition in the language of homo-morphisms of signed graphs, we show that there are signed planar simple graphs which are not 4-colorable. That is to say: there is a signed planar simple graph which is neither (2, 0)-colorable, nor (1, 1)-colorable, nor (0, 2)-colorable. That every signed planar simple graph is (2, 0)-colorable was the subject of a conjecture by Máčajová, Raspaud andŠkoviera which was recently disproved by Kardoš and Narboni using a dual notion. We provide a direct approach and a short proof. That every signed planar simple graph is (1, 1)-colorable is a recent conjecture of Jiang and Zhu which we disprove in this work. Noting that (0, 2)-coloring of (G, σ) is the same as (2, 0)-coloring of (G, −σ), this proves the existence of a signed planar simple graph whose complex chromatic number is larger than 4. Further developing the homomorphism approach, and as an analogue of the 5-color theorem, we find three minimal signed graphs each on three vertices, without a K ± 1 (a vertex with both a positive and a negative loop) and each having the property that admits a homomorphism from every signed planar simple graph. Finally we identify several other problems of high interest in colorings and homomorphisms of signed planar simple graphs.
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hal-03000542 , version 1 (12-11-2020)

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  • HAL Id : hal-03000542 , version 1

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Reza Naserasr, Lan Anh Pham. Complex and homomorphic chromatic number of signed planar simple graphs. Graphs and Combinatorics, 2022. ⟨hal-03000542⟩
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