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## Section: New Results

### Advances in Graph Theory, Clone Theory and Multiple-Valued Logic

Participants : Quentin Brabant, Miguel Couceiro, Amedeo Napoli, François Pirot, Chedy Raïssi, Jean-Sébastien Sereni.

Keywords:

graph theory, graph colouring, extremal graph theory, chromatic number, multiple-valued logic, clone theory

#### Graph Theory

there exists a positive real $\alpha$ such that whenever $G$ is a planar graph and $A$ is a subset of its edges whose deletion makes $G$ triangle-free, there exists a subset ${A}^{\text{'}}$ of $A$ of size at least $\alpha |A|$ such that $G-\left(A\setminus {A}^{\text{'}}\right)$ is 3-colourable. This equivalence allows us to study restricted situations, where we can prove the statement to be true.

Still on graph colouring, we demonstrated [93] a conjecture by Zhang and Whu made in 2011, that for every positive integer $\Delta$, every ${K}_{4}$-minor-free graph with maximum degree $\Delta$ admits an equitable colouring with $k$ colours whenever $k\ge \frac{\Delta +3}{2}$. A key ingredient was to not use the discharging method and rather exploit decomposition trees of ${K}_{4}$-minor-free graphs.

We also considered [88] distance colouring in graphs of maximum degree at most $d$ and how excluding one fixed cycle of length $\ell$ affects the number of colours required as $d\to \infty$. For vertex-colouring and $t\ge 1$, if any two distinct vertices connected by a path of at most $t$ edges are required to be coloured differently, then a reduction by a logarithmic (in $d$) factor against the trivial bound $O\left({d}^{t}\right)$ can be obtained by excluding an odd cycle length $\ell \ge +3t$ if $t$ is odd or by excluding an even cycle length $\ell \ge 2t+2$. For edge-colouring and $t\ge 2$, if any two distinct edges connected by a path of fewer than $t$ edges are required to be coloured differently, then excluding an even cycle length $\ell \ge 2t$ is sufficient for a logarithmic factor reduction. For $t\ge 2$, neither of the above statements are possible for other parity combinations of $\ell$ and $t$. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).

#### Multiple-Valued Logic and (Partial) Clone Theory

Clone theory was primarily motivated by the study of Boolean logic, and it currently constitutes a major subject in universal algebra, multiple-valued logic, and theoretical computer science. A clone on a set $A$ is a class of functions $f:{A}^{n}\to A$, $n\ge 1$, that contains all projections and that is closed under compositions. Clones on a set $A$ constitute a closure system, in fact, an algebraic lattice where meet is given by set-intersection. Clones on a 2-element set were completely classified by Emil Post. Since Post’s classification several studies on clone theory have appeared and many variants and generalizations have been proposed.

As a closure system, clones can be specified within a Galois framework, namely, through the well known Pol-Inv Galois connection by the polarity between functions and the relations they preserve. This Galois connection became the main tool in several studies, in particular, in the classification of the complexity classes of CSPs (“Constraint Satisfaction Problems”) [92]. Another, rather surprisingly, application of this Galois framework led to the description of of analogy-preserving Boolean classifiers [4].

Similarly, clones of partial functions (i.e., functions $f:D\to A$ for $D\subseteq {A}^{n}$) can be described by the relations its members preserve. Unlike the lattice of clones, the lattice of partial clones is of continuum cardinality even in the case of 2-element underlying sets. This shows that a complete description of this lattice is hard to attain. However, many efforts have been made towards local descriptions of this lattice, for instance, concerning the classification of its intervals that has entailed a long lasting open problem. This was settled [20] in the form of a dichotomy theorem showing that such intervals are either finite or of continuum cardinality, and we presented precise descriptions of the structure some challenging intervals in [21]. Further developments and related problems were also tackled in [11], [24], [40], [39].