L-orthogonality in Daugavet centers and narrow operators

We study the presence of L-orthogonal elements in connection with Daugavet centers and narrow operators. We prove that if $\mathrm{dens}(Y)?{\omega }_1$ and $G:X?Y$ is a Daugavet center with separable range then, for every non-empty $w^{?}$-open subset W of $B_{X^{??}}$, it follows that $G^{??}(W)$ contains some L-orthogonal to Y. In the context of narrow operators, we show that if X is separable and $T:X?Y$ is a narrow operator, then given $y\in B_X$ and any non-empty $w^{?}$-open subset W of $B_{X^{??}}$ then W contains some L-orthogonal u so that $T^{??}(u)=T(y)$. In the particular case that $T^{?}(Y^{?})$ is separable, we extend the previous result to $\mathrm{dens}(X)={\omega }_1$. Finally, we prove that none of the previous results holds in larger density characters (in particular, a counterexample is shown for ${\omega }_2$ under the assumption $2^c={\omega }_2$).     

Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }{X}_{+}$, where $X_{+}=X??$, for a smooth scheme $X\in {\mathrm{Sm}}_k$ over an infinite perfect field k, is computed.The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $({\mathbb{A}}^l\times X)/(({\mathbb{A}}^l?0)\times X)$, $X\in {\mathrm{Sm}}_k$, is one of ingredients in the theory. In the article we extend this result to the case of a pair $(X,U)$ given by a smooth affine variety X over k and an open subscheme $U?X$.The result gives an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }(X_{+}/U_{+})$ of the quotient-sheaf $X_{+}/U_{+}$.     

Point partition numbers: Decomposable and indecomposable critical graphs

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number ${\chi }_t(G)$ is the least integer k for which G admits a coloring with k colors such that each color class induces a $(t?1)$-degenerate subgraph of G. So ${\chi }_1$ is the chromatic number and ${\chi }_2$ is the point arboricity. The point partition number ${\chi }_t$ with $t\ge 1$ was introduced by Lick and White. A graph G is called ${\chi }_t$-critical if every proper subgraph H of G satisfies ${\chi }_t(H)<{\chi }_t(G)$. In this paper we prove that if G is a ${\chi }_t$-critical graph whose order satisfies $|G|\le 2{\chi }_t(G)?2$, then G can be obtained from two non-empty disjoint subgraphs $G_1$ and $G_2$ by adding t edges between any pair $u,v$ of vertices with $u\in V(G_1)$ and $v\in V(G_2)$. Based on this result we establish the minimum number of edges possible in a ${\chi }_t$-critical graph G of order n and with ${\chi }_t(G)=k$, provided that $n\le 2k?1$ and t is even. For $t=1$ the corresponding two results were obtained in 1963 by Tibor Gallai.     

On the equational graphs over finite fields

In this paper, we generalize the notion of functional graph. Specifically, given an equation $E(X,Y)=0$ with variables X and Y over a finite field ${\mathbb{F}}_q$ of odd characteristic, we define a digraph by choosing the elements in ${\mathbb{F}}_q$ as vertices and drawing an edge from x to y if and only if $E(x,y)=0$. We call this graph as equational graph. In this paper, we study the equational graph when choosing $E(X,Y)=(Y^2-f(X))(\lambda Y^2-f(X))$ with $f(X)$ a polynomial over ${\mathbb{F}}_q$ and λ a non-square element in ${\mathbb{F}}_q$. We show that if f is a permutation polynomial over ${\mathbb{F}}_q$, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials f of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.     

On the chromatic number of some P5-free graphs

Let G be a graph. We say that G is perfectly divisible if for each induced subgraph H of G, $V(H)$ can be partitioned into A and B such that $H[A]$ is perfect and $\omega (H[B])<\omega (H)$. We use $P_t$ and $C_t$ to denote a path and a cycle on t vertices, respectively. For two disjoint graphs $F_1$ and $F_2$, we use $F_1\cup F_2$ to denote the graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)$, and use $F_1+F_2$ to denote the graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)\cup \{xy|x\in V(F_1)\text{ and }y\in V(F_2)\}$. In this paper, we prove that (i) $(P_5,C_5,K_{2,3})$-free graphs are perfectly divisible, (ii) $\chi (G)\le 2{\omega }^2(G)?\omega (G)?3$ if G is $(P_5,K_{2,3})$-free with $\omega (G)\ge 2$, (iii) $\chi (G)\le \frac{3}{2}({\omega }^2(G)?\omega (G))$ if G is $(P_5,K_1+2K_2)$-free, and (iv) $\chi (G)\le 3\omega (G)+11$ if G is $(P_5,K_1+(K_1\cup K_3))$-free.