Spectrally degenerate graphs: Hereditary case

It is well known that the spectral radius of a tree whose maximum degree is D cannot exceed 2sqrt{D-1}. Similar upper bound holds for arbitrary planar graphs, whose spectral radius cannot exceed sqrt{8D}+10, and more generally, for all d-degenerate graphs, where the corresponding upper bound is sqrt{4dD}. Following this, we say that a graph G is spectrally d-degenerate if every subgraph H of G has spectral radius at most sqrt{d.Delta(H)}. In this paper we derive a rough converse of the above-mentioned results by proving that each spectrally d-degenerate graph G contains a vertex whose degree is at most 4dlog_2(D/d) (if D>=2d). It is shown that the dependence on D in this upper bound cannot be eliminated, as long as the dependence on d is subexponential. It is also proved that the problem of deciding if a graph is spectrally d-degenerate is co-NP-complete.Comment: Updated after reviewer comments. 14 pages, no figure

C4-free subgraphs with large average degree

Motivated by a longstanding conjecture of Thomassen, we study how large the average degree of a graph needs to be to imply that it contains a C4-free subgraph with average degree at least t. Kühn and Osthus showed that an average degree bound which is double exponential in t is sufficient. We give a short proof of this bound, before reducing it to a single exponential. That is, we show that any graph G with average degree at least 2ct2log t (for some constant c > 0) contains a C4-free subgraph with average degree at least t. Finally, we give a construction which improves the lower bound for this problem, showing that this initial average degree must be at least t3−o(1)

C4-free subgraphs with large average degree

Motivated by a longstanding conjecture of Thomassen, we study how large the average degree of a graph needs to be to imply that it contains a $C_4$-free subgraph with average degree at least $t$. K\"uhn and Osthus showed that an average degree bound which is double exponential in t is sufficient. We give a short proof of this bound, before reducing it to a single exponential. That is, we show that any graph $G$ with average degree at least $2^{ct^2\log t}$ (for some constant $c>0$) contains a $C_4$-free subgraph with average degree at least $t$. Finally, we give a construction which improves the lower bound for this problem, showing that this initial average degree must be at least $t^{3-o(1)}$

Nearly-linear monotone paths in edge-ordered graphs

How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chv´atal and Koml´os in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n 2/3−o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n 1−o(1

Nearly-linear monotone paths in edge-ordered graphs

How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n^2/3−o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n^1−o(1)