306 research outputs found

### Hadwiger's conjecture for graphs with forbidden holes

Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the
largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor.
A hole in $G$ is an induced cycle of length at least four. Hadwiger's
Conjecture from 1943 states that for every graph $G$, $h(G)\ge \chi(G)$, where
$\chi(G)$ denotes the chromatic number of $G$. In this paper we establish more
evidence for Hadwiger's conjecture by showing that if a graph $G$ with
independence number $\alpha(G)\ge3$ has no hole of length between $4$ and
$2\alpha(G)-1$, then $h(G)\ge\chi(G)$. We also prove that if a graph $G$ with
independence number $\alpha(G)\ge2$ has no hole of length between $4$ and
$2\alpha(G)$, then $G$ contains an odd clique minor of size $\chi(G)$, that is,
such a graph $G$ satisfies the odd Hadwiger's conjecture

### Saturation numbers for Ramsey-minimal graphs

Given graphs $H_1, \dots, H_t$, a graph $G$ is $(H_1, \dots,
H_t)$-Ramsey-minimal if every $t$-coloring of the edges of $G$ contains a
monochromatic $H_i$ in color $i$ for some $i\in\{1, \dots, t\}$, but any proper
subgraph of $G$ does not possess this property. We define
$\mathcal{R}_{\min}(H_1, \dots, H_t)$ to be the family of $(H_1, \dots,
H_t)$-Ramsey-minimal graphs. A graph $G$ is \dfn{$\mathcal{R}_{\min}(H_1,
\dots, H_t)$-saturated} if no element of $\mathcal{R}_{\min}(H_1, \dots, H_t)$
is a subgraph of $G$, but for any edge $e$ in $\overline{G}$, some element of
$\mathcal{R}_{\min}(H_1, \dots, H_t)$ is a subgraph of $G + e$. We define
$sat(n, \mathcal{R}_{\min}(H_1, \dots, H_t))$ to be the minimum number of edges
over all $\mathcal{R}_{\min}(H_1, \dots, H_t)$-saturated graphs on $n$
vertices. In 1987, Hanson and Toft conjectured that $sat(n,
\mathcal{R}_{\min}(K_{k_1}, \dots, K_{k_t}) )= (r - 2)(n - r + 2)+\binom{r -
2}{2}$ for $n \ge r$, where $r=r(K_{k_1}, \dots, K_{k_t})$ is the classical
Ramsey number for complete graphs. The first non-trivial case of Hanson and
Toft's conjecture for sufficiently large $n$ was setteled in 2011, and is so
far the only settled case. Motivated by Hanson and Toft's conjecture, we study
the minimum number of edges over all $\mathcal{R}_{\min}(K_3,
\mathcal{T}_k)$-saturated graphs on $n$ vertices, where $\mathcal{T}_k$ is the
family of all trees on $k$ vertices. We show that for $n \ge 18$, $sat(n,
\mathcal{R}_{\min}(K_3, \mathcal{T}_4)) =\lfloor {5n}/{2}\rfloor$. For $k \ge
5$ and $n \ge 2k + (\lceil k/2 \rceil +1) \lceil k/2 \rceil -2$, we obtain an
asymptotic bound for $sat(n, \mathcal{R}_{\min}(K_3, \mathcal{T}_k))$.Comment: to appear in Discrete Mathematic

### A Note on Weighted Rooted Trees

Let $T$ be a tree rooted at $r$. Two vertices of $T$ are related if one is a
descendant of the other; otherwise, they are unrelated. Two subsets $A$ and $B$
of $V(T)$ are unrelated if, for any $a\in A$ and $b\in B$, $a$ and $b$ are
unrelated. Let $\omega$ be a nonnegative weight function defined on $V(T)$ with
$\sum_{v\in V(T)}\omega(v)=1$. In this note, we prove that either there is an
$(r, u)$-path $P$ with $\sum_{v\in V(P)}\omega(v)\ge \frac13$ for some $u\in
V(T)$, or there exist unrelated sets $A, B\subseteq V(T)$ such that $\sum_{a\in
A }\omega(a)\ge \frac13$ and $\sum_{b\in B }\omega(b)\ge \frac13$. The bound
$\frac13$ is tight. This answers a question posed in a very recent paper of
Bonamy, Bousquet and Thomass\'e

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