Planar graphs without specific cycles are 2-degenerate

A graph G is k-degenerate if each subgraph of G has a vertex of degree at most k. It is known that every simple planar graph with girth 6, or equivalently without 3-, 4-, and 5-cycles, is 2-degenerate. In this work, we investigate for which k every planar graph without 4-, 6-, … , and 2k-cycles is 2-degenerate. We determine that k is 5 and the result is tight since the truncated dodecahedral graph is a 3-regular planar graph without 4-, 6-, and 8-cycles. As a related result, we also show that every planar graph without 4-, 6-, 9-, and 10-cycles is 2-degenerate.     

The Decomposition Dimension of Graphs

 For an ordered k-decomposition ? = {G ₁, G ₂,…,G _k } of a connected graph G and an edge e of G, the ?-representation of e is the k-tuple r(e|?) = (d(e, G ₁), d(e, G ₂),…,d(e, G _k )), where d(e, G _i ) is the distance from e to G _i . A decomposition ? is resolving if every two distinct edges of G have distinct representations. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dec(G). It is shown that for every two positive integers k and n≥ 2, there exists a tree T of order n with dec(T) = k. It is also shown that dec(G) ≤n for every graph G of order n≥ 3 and that dec(K _n ) ≤⌊(2n + 5)/3⌋ for n≥ 3. Received: June 17, 1998 Final version received: August 10, 1999     

Total Colorings Of Degenerate Graphs

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is s-degenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤s. We prove that an s-degenerate graph G has a total coloring with Δ+1 colors if the maximum degree Δ of G is sufficiently large, say Δ≥4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a lineartime algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.     

Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex υ is called the weighted degree of υ. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2‐connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2‐connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

A sufficient degree condition for a graph to contain all trees of size <Emphasis Type="Italic">k</Emphasis>

The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d _G (x) + d _G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d _G (u) ≥ k.     

Circular perfect graphs

For 1 ≤ d ≤ k, let K_k/d be the graph with vertices 0, 1, …, k ? 1, in which i ～j if d ≤ |i ? j| ≤ k ? d. The circular chromatic number χ_c(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to K_k/d. The circular clique number ω_c(G) of G is the maximum of those k/d for which K_k/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χ_c(H) = ω_c(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, N_G[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, N_G[x] is a perfect graph and G ? N[x] is a bipartite graph with no induced P₅ (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj?os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph K_k/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005     

Two remarks on the Burr–Erdős conjecture

The Ramsey number r(H) of a graph H is the minimum positive integer N such that every two-coloring of the edges of the complete graph K_N on N vertices contains a monochromatic copy of H. A graph H is d-degenerate if every subgraph of H has minimum degree at most d. Burr and Erdős in 1975 conjectured that for each positive integer d there is a constant c_d such that r(H)≤c_dn for every d-degenerate graph H on n vertices. We show that for such graphs , improving on an earlier bound of Kostochka and Sudakov. We also study Ramsey numbers of random graphs, showing that for d fixed, almost surely the random graph G(n,d/n) has Ramsey number linear in n. For random bipartite graphs, our proof gives nearly tight bounds.     

The Erdös-Jacobson-Lehel conjecture on potentiallyP k-graphic sequence is true

A variation in the classical Turan extrernal problem is studied. A simple graphG of ordern is said to have propertyPk if it contains a clique of sizek+1 as its subgraph. Ann-term nonincreasing nonnegative integer sequence π=(d₁, d₂,⋯, d₂) is said to be graphic if it is the degree sequence of a simple graphG of ordern and such a graphG is referred to as a realization of π. A graphic sequence π is said to be potentiallyP _k-graphic if it has a realizationG having propertyP _k . The problem: determine the smallest positive even number σ(k, n) such that everyn-term graphic sequence π=(d₁, d₂,…, d₂) without zero terms and with degree sum σ(π)=(d ₁+d ₂+ …+d ₂) at least σ(k,n) is potentially P_k-graphic has been proved positive. Project supported by the National Natural Science Foundation of China (Grant No. 19671077) and the Doctoral Program Foundation of National Education Department of China.     

On low bound of degree sequences of spanning trees in K-edge-connected graphs

A graph G = (V, E) is k-edge-connected if for any subset E′ ⊆ E,|E′| < k, G − E′ is connected. A d_k-tree T of a connected graph G = (V, E) is a spanning tree satisfying that ∀v ∈ V, d_T(v) ≤ + α, where [·] is a lower integer form and α depends on k. We show that every k-edge-connected graph with k ≥ 2, has a d_k-tree, and α = 1 for k = 2, α = 2 for k ≥ 3. © 1998 John Wiley & Sons, Inc. J Graph Theory 28: 87–95, 1998     

k-ordered Hamiltonian graphs

A hamiltonian graph G of order n is k-ordered, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices of G, there exists a hamiltonian cycle that encounters v₁, v₂, …, v_k in this order. Theorems by Dirac and Ore, presenting sufficient conditions for a graph to be hamiltonian, are generalized to k-ordered hamiltonian graphs. The existence of k-ordered graphs with small maximum degree is investigated; in particular, a family of 4-regular 4-ordered graphs is described. A graph G of order n ≥ 3 is k-hamiltonian-connected, 2 ≤ k ≤ n, if for every sequence v₁, v₂, …, v_k of k distinct vertices, G contains a v₁-v_k hamiltonian path that encounters v₁, v₂,…, v_k in this order. It is shown that for k ≥ 3, every (k + 1)-hamiltonian-connected graph is k-ordered and a result of Ore on hamiltonian-connected graphs is generalized to k-hamiltonian-connected graphs. © 1997 John Wiley & Sons, Inc.     

Removable edges in cycles of a <Emphasis Type="Italic">k</Emphasis>-connected graph

An edge e of a k-connected graph G is said to be a removable edge if G ⊖ e is still k-connected, where G ⊖ e denotes the graph obtained from G by deleting e to get G − e, and for any end vertex of e with degree k − 1 in G − e, say x, delete x, and then add edges between any pair of non-adjacent vertices in N _G−e (x). The existence of removable edges of k-connected graphs and some properties of 3-connected graphs and 4-connected graphs have been investigated. In the present paper, we investigate some properties of k-connected graphs and study the distribution of removable edges on a cycle in a k-connected graph (k ≥ 4).     

Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

A graph G is (k,0)‐colorable if its vertices can be partitioned into subsets V₁ and V₂ such that in G[V₁] every vertex has degree at most k, while G[V₂] is edgeless. For every integer k?0, we prove that every graph with the maximum average degree smaller than (3k+4)/(k+2) is (k,0)‐colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)‐colorable. On the other hand, we construct planar graphs with girth 6 that are not (k,0)‐colorable for arbitrarily large k. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:83–93, 2010     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)     

A characterization of maximal non-k-factor-critical graphs

A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but G+e is k-factor-critical for every missing edge e∉E(G). A connected graph G with a perfect matching on 2n vertices is k-extendable, for 1?k?n-1, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal non-k-extendable if G is not k-extendable but G+e is k-extendable for every missing edge e∉E(G) . A connected bipartite graph G with a bipartitioning set (X,Y) such that |X|=|Y|=n is maximal non-k-extendable bipartite if G is not k-extendable but G+xy is k-extendable for any edge xy∉E(G) with x∈X and y∈Y. A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.     

The relaxed game chromatic index of k-degenerate graphs

The (r,d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r,d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with Δ(G)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ+k-1 and d?2k²+4k.     

On 2-factors with given properties in claw-free graphs

Let G be a claw-free graph such that (i) k(G) 3 2k(G) \geq 2, (ii) $|V (G)| \geq 8$|V (G)| \geq 8 and (iii) d(G) 3 4\delta(G) \geq 4. For every pair of edges e₁, e₂ of G the graph G^* = G - {e₁, e₂}G^* = G - \{e_1, e_2\} has a 2-factor.     

Pancyclic graphs and a conjecture of Bondy and Chvátal

A p-vertex graph is called pancyclic if it contains cycles of every length l, 3 ≤ l ≤ p. In this paper we prove the following conjecture of Bondy and Chvátal: If a graph G has vertex degree sequence d₁ ≤ d₂ ≤ … ≤ d_ν, and if $\text{d}{}_{\mathrm{k}}\text{≤ k <}\text{p}\text{2}$ implies d_ν?k ≥ p ? k, then G is pancyclic or bipartite.     

[a,b]-factorization of a graph

Let a and b be integers such that 0 ? a ? b. Then a graph G is called an [a, b]-graph if a ? d_G(x) ? b for every x ? V(G), and an [a, b]-factor of a graph is defined to be its spanning subgraph F such that a ? d_F(x) ? b for every vertex x, where d_G(x) and d_F(x) denote the degrees of x in G and F, respectively. If the edges of a graph can be decomposed into [a.b]-factors then we say that the graph is [2a, 2a]-factorable. We prove the following two theorems: (i) a graph G is [2a, 2b)-factorable if and only if G is a [2am,2bm]-graph for some integer m, and (ii) every [8m + 2k, 10m + 2k]-graph is [1,2]-factorable.     

Degree conditions for k‐ordered hamiltonian graphs

For a positive integer k, a graph G is k-ordered hamiltonian if for every ordered sequence of k vertices there is a hamiltonian cycle that encounters the vertices of the sequence in the given order. It is shown that if G is a graph of order n with 3 ≤ k ≤ n/2, and deg(u) + deg(v) ≥ n + (3k − 9)/2 for every pair u, v of nonadjacent vertices of G, then G is k-ordered hamiltonian. Minimum degree conditions are also given for k-ordered hamiltonicity. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 199–210, 2003     

Edge disjoint Hamilton cycles in graphs

Let G be a graph of order n and k ≥ 0 an integer. It is conjectured in [8] that if for any two vertices u and v of a 2(k + 1)‐connected graph G,d _G (u,v) = 2 implies that max{d(u;G), d(v;G)} ≥ (n/2) + 2k, then G has k + 1 edge disjoint Hamilton cycles. This conjecture is true for k = 0, 1 (see cf. [3] and [8]). It will be proved in this paper that the conjecture is true for every integer k ≥ 0. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 8–20, 2000