306 research outputs found

    Hadwiger's conjecture for graphs with forbidden holes

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    Given a graph GG, the Hadwiger number of GG, denoted by h(G)h(G), is the largest integer kk such that GG contains the complete graph KkK_k as a minor. A hole in GG is an induced cycle of length at least four. Hadwiger's Conjecture from 1943 states that for every graph GG, h(G)β‰₯Ο‡(G)h(G)\ge \chi(G), where Ο‡(G)\chi(G) denotes the chromatic number of GG. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph GG with independence number Ξ±(G)β‰₯3\alpha(G)\ge3 has no hole of length between 44 and 2Ξ±(G)βˆ’12\alpha(G)-1, then h(G)β‰₯Ο‡(G)h(G)\ge\chi(G). We also prove that if a graph GG with independence number Ξ±(G)β‰₯2\alpha(G)\ge2 has no hole of length between 44 and 2Ξ±(G)2\alpha(G), then GG contains an odd clique minor of size Ο‡(G)\chi(G), that is, such a graph GG satisfies the odd Hadwiger's conjecture

    Saturation numbers for Ramsey-minimal graphs

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    Given graphs H1,…,HtH_1, \dots, H_t, a graph GG is (H1,…,Ht)(H_1, \dots, H_t)-Ramsey-minimal if every tt-coloring of the edges of GG contains a monochromatic HiH_i in color ii for some i∈{1,…,t}i\in\{1, \dots, t\}, but any proper subgraph of GG does not possess this property. We define Rmin⁑(H1,…,Ht)\mathcal{R}_{\min}(H_1, \dots, H_t) to be the family of (H1,…,Ht)(H_1, \dots, H_t)-Ramsey-minimal graphs. A graph GG is \dfn{Rmin⁑(H1,…,Ht)\mathcal{R}_{\min}(H_1, \dots, H_t)-saturated} if no element of Rmin⁑(H1,…,Ht)\mathcal{R}_{\min}(H_1, \dots, H_t) is a subgraph of GG, but for any edge ee in Gβ€Ύ\overline{G}, some element of Rmin⁑(H1,…,Ht)\mathcal{R}_{\min}(H_1, \dots, H_t) is a subgraph of G+eG + e. We define sat(n,Rmin⁑(H1,…,Ht))sat(n, \mathcal{R}_{\min}(H_1, \dots, H_t)) to be the minimum number of edges over all Rmin⁑(H1,…,Ht)\mathcal{R}_{\min}(H_1, \dots, H_t)-saturated graphs on nn vertices. In 1987, Hanson and Toft conjectured that sat(n,Rmin⁑(Kk1,…,Kkt))=(rβˆ’2)(nβˆ’r+2)+(rβˆ’22)sat(n, \mathcal{R}_{\min}(K_{k_1}, \dots, K_{k_t}) )= (r - 2)(n - r + 2)+\binom{r - 2}{2} for nβ‰₯rn \ge r, where r=r(Kk1,…,Kkt)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 nn 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 Rmin⁑(K3,Tk)\mathcal{R}_{\min}(K_3, \mathcal{T}_k)-saturated graphs on nn vertices, where Tk\mathcal{T}_k is the family of all trees on kk vertices. We show that for nβ‰₯18n \ge 18, sat(n,Rmin⁑(K3,T4))=⌊5n/2βŒ‹sat(n, \mathcal{R}_{\min}(K_3, \mathcal{T}_4)) =\lfloor {5n}/{2}\rfloor. For kβ‰₯5k \ge 5 and nβ‰₯2k+(⌈k/2βŒ‰+1)⌈k/2βŒ‰βˆ’2n \ge 2k + (\lceil k/2 \rceil +1) \lceil k/2 \rceil -2, we obtain an asymptotic bound for sat(n,Rmin⁑(K3,Tk))sat(n, \mathcal{R}_{\min}(K_3, \mathcal{T}_k)).Comment: to appear in Discrete Mathematic

    A Note on Weighted Rooted Trees

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    Let TT be a tree rooted at rr. Two vertices of TT are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets AA and BB of V(T)V(T) are unrelated if, for any a∈Aa\in A and b∈Bb\in B, aa and bb are unrelated. Let Ο‰\omega be a nonnegative weight function defined on V(T)V(T) with βˆ‘v∈V(T)Ο‰(v)=1\sum_{v\in V(T)}\omega(v)=1. In this note, we prove that either there is an (r,u)(r, u)-path PP with βˆ‘v∈V(P)Ο‰(v)β‰₯13\sum_{v\in V(P)}\omega(v)\ge \frac13 for some u∈V(T)u\in V(T), or there exist unrelated sets A,BβŠ†V(T)A, B\subseteq V(T) such that βˆ‘a∈AΟ‰(a)β‰₯13\sum_{a\in A }\omega(a)\ge \frac13 and βˆ‘b∈BΟ‰(b)β‰₯13\sum_{b\in B }\omega(b)\ge \frac13. The bound 13\frac13 is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomass\'e
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