[Paper Review] Degree powers in graphs with forbidden even cycle

Vladimir Nikiforov|ArXiv.org|Mar 31, 2009

Limits and Structures in Graph Theory4 references3 citations

TL;DR

This paper resolves a conjecture by Caro and Yuster on the maximum sum of $ p $-th powers of vertex degrees in graphs without an even cycle $ C_{2k+2} $. Using a novel sufficient condition for long paths in graphs, the authors prove that this maximum sum is asymptotically $ kn^p(1 + o(1)) $, establishing a tight upper bound via degree sequence analysis and spectral graph theory techniques.

ABSTRACT

We prove a conjecture of Yuster and Caro about the sum of the p-powers of the degrees of a graph of order n without a specified even cycle. Our proof is based on a new sufficient condition for long paths, that may be useful in other applications as well.

Motivation & Objective

• To resolve a conjecture by Caro and Yuster on the maximum sum of $ p $-th powers of vertex degrees in graphs that exclude the even cycle $ C_{2k+2} $.
• To establish an asymptotically tight upper bound for the sum $ extstyleigsum_{u otin V(G)} d^p(u) $ in $ C_{2k+2} $-free graphs of order $ n $.
• To develop and apply a new sufficient condition for the existence of long paths with both endpoints in a given vertex set, which is central to the proof.
• To derive spectral and degree-based inequalities that constrain the structure of $ C_{2k+2} $-free graphs, leading to the main result.
• To provide a clean, general bound that applies for all $ p o ext{infty} $, extending beyond quadratic degree sums.

Proposed method

• Introduces a new sufficient condition (Lemma 1) for the existence of long paths in graphs partitioned into two sets $ A $ and $ B $, based on edge counts involving $ A $ and between $ A $ and $ B $.
• Applies Lemma 1 to the neighborhood of a vertex $ u $, partitioning $ G - u $ into $ A = ext{neighbors of } u $ and $ B = ext{non-neighbors} $, to avoid $ A $-paths of order $ 2k+1 $.
• Uses the absence of such paths to derive an inequality on the sum of degrees of neighbors of $ u $, leading to $ extstyleigsum_{v otin ext{N}(u)} d(v) ext{ bounded by } kd(u) + k(n-1) $.
• Sums over all vertices $ u $, transforming the double sum into $ extstyleigsum_{v} d^2(v) $, and applies the AM-QM inequality to bound the number of edges.
• Extends the quadratic bound to higher powers $ p o ext{infty} $ by using the edge bound $ m = O(n^{3/2}) $, leading to $ extstyleigsum d^p(u) ext{ bounded by } kn^p + O(n^{p-1/2}) $.
• Employs spectral graph theory to derive a bound on the largest eigenvalue $ u $ of the adjacency matrix: $ u^2 - k u ext{ bounded by } k(n-1) $, supporting the main result.

Experimental results

Research questions

• RQ1 What is the maximum possible value of $ extstyleigsum_{u otin V(G)} d^p(u) $ in a graph of order $ n $ that excludes the cycle $ C_{2k+2} $?
• RQ2 Can a new sufficient condition for long paths in graphs be used to derive tight bounds on degree power sums in $ C_{2k+2} $-free graphs?
• RQ3 Does the conjecture $ extstyleigsum d^p(u) = kn^p(1 + o(1)) $ hold for all $ p o ext{infty} $ in $ C_{2k+2} $-free graphs?
• RQ4 How do degree sequences and spectral properties constrain the structure of graphs without even cycles of length $ 2k+2 $?
• RQ5 Can the bound on the sum of squared degrees be extended to higher-degree powers using combinatorial and spectral techniques?

Key findings

• The maximum sum of $ p $-th powers of vertex degrees in a $ C_{2k+2} $-free graph of order $ n $ is asymptotically $ kn^p(1 + o(1)) $, confirming the Caro-Yuster conjecture.
• For $ p = 2 $, the sum $ extstyleigsum d^2(u) $ is bounded above by $ 2km + k(n-1)n $, where $ m $ is the number of edges, and this bound is tight up to lower-order terms.
• The number of edges $ m $ in a $ C_{2k+2} $-free graph satisfies $ m = O(n^{3/2}) $, derived from the quadratic bound and the AM-QM inequality.
• The bound $ extstyleigsum d^p(u) ext{ is } kn^p + O(n^{p-1/2}) $ holds for all $ p o ext{infty} $, showing that the degree $ p $-th power sum is tightly concentrated around $ kn^p $.
• The spectral bound $ u^2 - k u ext{ bounded by } k(n-1) $ holds for the largest eigenvalue $ u $ of the adjacency matrix in $ C_{2k+2} $-free graphs, supporting the main result.
• The new path existence lemma (Lemma 1) provides a powerful tool for analyzing extremal graphs without even cycles, and may be applicable in other extremal graph theory problems.