Since the diagonal lemma plays a key role in the proof of the main limitative theorems of logic, its proof could shed light on the very essence of these fundamental theorems. Yet the lemma is often characterized as one of those important logical results that lack an insightful explanatory proof. By making explicit that the well-known proof of the lemma is just a straightforward translation of the Grelling paradox into first-order arithmetic, the proof can be made completely transparent.

Gyorgy Sereny

It is conjectured that every fullerene graph is hamiltonian. Jendrol' and Owens proved [J. Math. Chem. 18 (1995), pp. 83--90] that every fullerene graph on n vertices has a cycle of length at least 4n/5. In this paper, we improve this bound to 5n/6-2/3.

D. Král' O. Pangrác J. -S. Sereni R. Skrekovski

Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most $\Delta+1+\varepsilon$. The conjecture was proven to be true when $\Delta=3$ or $\Delta$ is even. We settle the conjecture by proving it for the remaining cases.

Frantisek Kardos Daniel Kral Jean-Sebastien Sereni

We prove a "supersaturation-type" extension of both Sperner's Theorem (1928) and its generalization by Erdos (1945) to k-chains. Our result implies that a largest family whose size is x more than the size of a largest k-chain free family and that contains the minimum number of k-chains is the family formed by taking the middle (k-1) rows of the Boolean lattice and x elements from the k-th middle row. We prove our result using the symmetric chain decomposition method of de Bruijn, van Ebbenhorst Tengbergen, and Kruyswijk (1951).

Andrew P. Dove Jerrold R. Griggs Ross J. Kang Jean-Sébastien Sereni

This paper contributes to a programme initiated by the first author: `How much information about a graph is revealed in its Potts partition function?'. We show that the $W$-polynomial distinguishes non-isomorphic weighted trees of a \emph{good} family. The framework developed to do so also allows us to show that the $W$-polynomial distinguishes non-isomorphic caterpillars. This establishes Stanley's isomorphism conjecture for caterpillars, an extensively studied problem.

Martin Loebl Jean-Sébastien Sereni

Motivated by frequency assignment in office blocks, we study the chromatic number of the adjacency graph of $3$-dimensional parallelepiped arrangements. In the case each parallelepiped is within one floor, a direct application of the Four-Colour Theorem yields that the adjacency graph has chromatic number at most $8$. We provide an example of such an arrangement needing exactly $8$ colours. We also discuss bounds on the chromatic number of the adjacency graph of general arrangements of $3$-dimensional parallelepipeds according to geometrical measures of the parallelepipeds (side length, total surface or volume).

Stéphane Bessy Daniel Gonçalves Jean-Sébastien Sereni

Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: 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 $A'$ of $A$ of size at least $\alpha|A|$ such that $G-(A\setminus A')$ is $3$-colorable. This equivalence allows us to study restricted situations, where we can prove the statement to be true.

Zdeněk Dvořák Jean-Sébastien Sereni

On the occasion of the 80th anniversary of the first observation of Ce volume collapse in CeN a remembrance of the implications of that transcendent event is presented, along with a review of the knowledge of Ce physical properties available at that time. Coincident anniversary corresponds to the first proposal for Ce as a mix valence element, motivating to briefly review how the valence instability of Ce was investigated since that time.

Julian G. Sereni

We show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3. The previous best value, proved by Erdos, Faudree, Gould, Gy\'arf\'as, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.

Daniel Král' Chun-Hung Liu Jean-Sébastien Sereni Peter Whalen Zelealem Yilma

We demonstrate that for every positive integer $\Delta$, every K\_4-minor-free graph with maximum degree $\Delta$ admits an equitable coloring with k colors wherek $\ge$ ($\Delta$+3)/2. This bound is tight and confirms a conjecture by Zhang and Whu. We do not use the discharging method but rather exploit decomposition trees of K 4-minor-free graphs.

Rémi De Joannis de Verclos Jean-Sébastien Sereni