Almost-spanning subgraphs with bounded degree in dense graphs

• Authors: Yoshiyasu Ishigami
• Venue: European Journal of Combinatorics
• Year: 2002

We present two extensions of a theorem by Alon and Yuster (1992, Graphs Comb., 8, 95-102) that give degree conditions guaranteeing an almost-spanning subgraph isomorphic to a given graph. The first extension gives a sharp degree condition when the desired subgraph consists of small connected components (i.e., a packing of a host graph with small graphs), improving a theorem of Komlós (2000, Combinatorica, 20, 203-218). The second extension weakens the assumption of the desired subgraph in the Alon-Yuster theorem.Given a graph F, we write χ(F) and Δ(F) for the chromatic number and the maximum degree, respectively. We also denote by σr(F) the smallest possible colour class size in any proper r-vertex-colouring of F. The first theorem states that, for every Δ ≥ 1 and ε 0, there exists a µ 0 and an n0 such that the following holds. Let H = ∪i Hi be a non-empty graph such that |H| ≤ (1 - ε)n, Δ(H) ≤ Δ, and, for each Hi, |Hi| ≤ µn. Then every graph G with order n ≥ n0 and minimum degree δ(G) ≥ (1- (1 - σ)/(r - 1) - µ)n - contains a copy of H where r := maxi χ(Hi) and σ : = max{Σi σr (Hi)/|H|, ε}. The second theorem states that, for any r 1, Δ ≥ 0, and ε 0, there exists a µ 0 and an n0 such that for every graph G with order n ≥ n0 and δ(G) ≥ (1- (1- (1/r) - µ)n, one of the following two holds: • For any graph H with |H| ≤ (1 - ε)n, Δ(H) ≤ Δ, χ(H) ≤ r, and b(H) ≤ µn, G contains a copy of H (where b(H) denotes the bandwidth of H). • By deleting and adding at most εn2 edges and r vertices of G, G can be isomorphic to Kr(⌊n/r⌋) or K⌊n/r⌋ + Kr-2(⌊n/r⌋) + K⌊n/r⌋. The assumption of b(H) cannot be dropped.