For a simple graph $G$ with $n$ vertices, write its chromatic polynomial in the rising factorial basis as $$ \chi_G(x)=\sum_{i=0}^{n}(-1)^{n-i}c_i(G)\langle x\rangle_i,$$ where $ \langle x\rangle_i=x(x+1)\cdots(x+i-1).$ The associated $\tau$-polynomial $$ \tau_G(x)=\sum_{i=0}^{n}c_i(G)x^i $$ was defined and systematically investigated by Brenti in 1992. In this paper, we prove that if the $\tau$-polynomials of two vertex-disjoint simple graphs $G$ and $H$ have only real zeros, then the $\tau$-polynomial of their join $G\vee H$ has only real zeros. This settles a conjecture posed by Brenti, Royle and Wagner since 1994.
Let $P_t$ denote the induced path on $t$ vertices. Let $\omega(G)$ denote the maximum number of vertices in a clique of a graph $G$. Gy\'arf\'as (1987) proved that every $P_t$-free graph $G$ satisfies $\chi(G)\le(t-1)^{\omega(G)-1}$, and Gravier, Ho\`ang, and Maffray (2003) improved this to $\chi(G)\le (t-2)^{\omega(G)...
Let $r\ge2$, let $T_r$ denote the transitive tournament on $r$ vertices, and write $d_G^*(v):=\max\{d_G^+(v),d_G^-(v)\}$. We prove that if $r\mid n$ and an $n$-vertex digraph $G$ satisfies $d_G^*(x)+d_G^*(y)\ge 2(1-1/r)n-1$ for every $x\ne y \in V(G)$ with $xy \notin E(G)$, then $G$ has a $T_r$-factor, and the bound is...
Yu-jeong Chang, Shuo Wei, Jin Yan· 2 citations· ⚡1
Let $G$ be a finite simple graph on $[n]$ and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring $S = K[x_1,\dots,x_n]$. We give a combinatorial description, in terms of the structure of $G$, of the minimal generators of the symbolic Rees algebra $\mathcal{R}_s(I_c(G)) = \bigoplus_{k \geq 0} I_c(G)...
Antonino Ficarra, Somayeh Moradi, Y. Muta· 1 citation
Let $ G $ be a connected graph with $ n $ vertices and adjacency matrix $A(G)$. The critical polynomial $d_G(x_1, \ldots, x_n) $ is a degree-$n$ multivariate polynomial defined as the determinant of the matrix $M_G(x_1, \ldots, x_n) $, where \[M_G(x_1,\ldots,x_n) = \operatorname{Diag}(x_1, \ldots, x_n) - A(G).\] For an...
Ting-Ting Wang, Lu Lu· Electronic Journal of Combin...· 0 citations
For graphs $G,H$ and positive integers $r$ and $n$ we write $G^{\square n} \xrightarrow{r} H$ if every $r$-vertex-coloring of the Cartesian power $G^{\square n}$ of $G$ contains a monochromatic copy of $H$. Since chromatic number $\chi$ of $G^{\square n}$ is the same as $\chi(G)$, there is an $r$-vertex coloring of $G^...
N'ora Alm'asi, M. Axenovich, Arsenii Sagdeev· 1 citation
For an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erd\H{o}s and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)...
