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Register a new userTuzas Conjecture is Asymptotically Tight for Dense Graphs
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An old conjecture of Zs. Tuza says that for any graph $G$, the ratio of the minimum size, $tau_3(G)$, of a set of edges meeting all triangles to the maximum size, $ u_3(G)$, of an edge-disjoint triangle packing is at most 2. Here, disproving a conjecture of R. Yuster, we show that for any fixed, positive $alpha$ there are arbitrarily large graphs $G$ of positive density satisfying $tau_3(G)>(1-o(1))|G|/2$ and $ u_3(G)<(1+alpha)|G|/4$.
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Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including planar graphs. However, for dense graphs that are neither cliques nor 4-colorable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4.
Tuza (1981) conjectured that the size $tau(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $ u(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we present three results regarding Tuzas Conjecture. We verify it for graphs with treewidth at most $6$; we show that $tau(G)leq frac{3}{2}, u(G)$ for every planar triangulation $G$ different from $K_4$; and that $tau(G)leqfrac{9}{5}, u(G) + frac{1}{5}$ if $G$ is a maximal graph with treewidth 3. Our first result strengthens a result of Tuza, implying that $tau(G) leq 2, u(G)$ for every $K_8$-free chordal graph $G$.
The Shannon lower bound is one of the few lower bounds on the rate-distortion function that holds for a large class of sources. In this paper, it is demonstrated that its gap to the rate-distortion function vanishes as the allowed distortion tends to zero for all sources having a finite differential entropy and whose integer part is finite. Conversely, it is demonstrated that if the integer part of the source has an infinite entropy, then its rate-distortion function is infinite for every finite distortion. Consequently, the Shannon lower bound provides an asymptotically tight bound on the rate-distortion function if, and only if, the integer part of the source has a finite entropy.
We apply the Discharging Method to prove the 1,2,3-Conjecture and the 1,2-Conjecture for graphs with maximum average degree less than 8/3. Stronger results on these conjectures have been proved, but this is the first application of discharging to them, and the structure theorems and reducibility results are of independent interest.
A $k$-uniform tight cycle is a $k$-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size $k$ formed by $k$ consecutive vertices in the ordering. We prove that every red-blue edge-coloured $K_n^{(4)}$ contains a red and a blue tight cycle that are vertex-disjoint and together cover $n-o(n)$ vertices. Moreover, we prove that every red-blue edge-coloured $K_n^{(5)}$ contains four monochromatic tight cycles that are vertex-disjoint and together cover $n-o(n)$ vertices.
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