On Vizing’s bound for the chromatic index of a multigraph

Two of the basic results on edge coloring are Vizing’s Theorem [V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25-30 (in Russian); V.G. Vizing, The chromatic class of a multigraph, Kibernetika (Kiev) 3 (1965) 29-39 (in Russian). English translation in Cybernetics 1 32-41], which states that the chromatic index χ^′(G) of a (multi)graph G with maximum degree Δ(G) and maximum multiplicity μ(G) satisfies Δ(G)≤χ^′(G)≤Δ(G)+μ(G), and Holyer’s Theorem [I. Holyer, The NP-completeness of edge-colouring, SIAM J. Comput. 10 (1981) 718-720], which states that the problem of determining the chromatic index of even a simple graph is NP-hard. Hence, a good characterization of those graphs for which Vizing’s upper bound is attained seems to be unlikely. On the other hand, Vizing noticed in the first two above-cited references that the upper bound cannot be attained if Δ(G)=2μ(G)+1≥5. In this paper we discuss the problem: For which values Δ,μ does there exist a graph G satisfying Δ(G)=Δ, μ(G)=μ, and χ^′(G)=Δ+μ.     

On Seymour’s strengthening of Hadwiger’s conjecture for graphs with certain forbidden subgraphs

Let H be a set of graphs. A graph is called H-free if it does not contain a copy of a member of H as an induced subgraph. If H is a graph then G is called H-free if it is {H}-free. Plummer, Stiebitz, and Toft proved that, for every -free graph H on at most four vertices, every -free graph G has a collection of ⌈|V(G)|/2⌉ many pairwise adjacent vertices and edges (where a vertexvand an edgeeare adjacent if v is disjoint from the set V(e) of endvertices of e and adjacent to some vertex of V(e), and two edgeseandfare adjacent if V(e) and V(f) are disjoint and some vertex of V(e) is adjacent to some vertex of V(f)). Here we generalize this statement to -free graphs H on at most five vertices.     

Graham’s pebbling conjecture on product of thorn graphs of complete graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p₁,p₂,…,p_n be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p₁,p₂,…,p_n, is obtained by attaching p_i new vertices of degree 1 to the vertex u_i of the graph G, i=1,2,…,n. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). We show that Graham’s conjecture holds true for a thorn graph of the complete graph with every by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every .     

The edge version of Hadwiger’s conjecture

A well known conjecture of Hadwiger asserts that K_n+1 is the only minor minimal graph of chromatic number greater than n. In this paper, all minor minimal graphs of chromatic index greater than n are determined.     

The Erd s-Sós conjecture for graphs of girth 5

We prove that every graph of girth at least 5 with minimum degree δ k/2 contains every tree with k edges, whose maximum degree does not exceed the maximum degree of the graph. An immediate consequence is that the famous Erd s-Sós Conjecture, saying that every graph of order n with more than n(k − 1)/2 edges contains every tree with k edges, is true for graphs of girth at least 5.     

Graham’s pebbling conjecture on products of many cycles

A pebbling move on a connected graph G consists of removing two pebbles from some vertex and adding one pebble to an adjacent vertex. We define f_t(G) as the smallest number such that whenever f_t(G) pebbles are on G, we can move t pebbles to any specified, but arbitrary vertex. Graham conjectured that f₁(G×H)≤f₁(G)f₁(H) for any connected G and H. We define the α-pebbling number α(G) and prove that α(C_pj×?×C_p2×C_p1×G)≤α(C_pj)?α(C_p2)α(C_p1)α(G) when none of the cycles is C₅, and G satisfies one more criterion. We also apply this result with G=C₅×C₅ by showing that C₅×C₅ satisfies Chung’s two-pebbling property, and establishing bounds for f_t(C₅×C₅).     

Hadwiger's conjecture for quasi‐line graphs

A graph G is a quasi‐line graph if for every vertex v ∈ V(G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. Hadwiger's conjecture states that if a graph G is not t‐colorable then it contains K_{t + 1} as a minor. This conjecture has been proved for line graphs by Reed and Seymour. We extend their result to all quasi‐line graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 17–33, 2008     

Gallai’s path decomposition conjecture for triangle-free planar graphs

A path decomposition of a graph $G$ is a collection of edge-disjoint paths of $G$ that covers the edge set of $G$. Gallai (1968) conjectured that every connected graph on $n$ vertices admits a path decomposition of cardinality at most $?(n+1)∕2?$. Gallai’s Conjecture has been verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex with even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex with odd degree. Recently, Bonamy and Perrett (2016) verified Gallai’s Conjecture for graphs with maximum degree at most 5, and Botler et al. (2017) verified it for graphs with treewidth at most 3. In this paper, we verify Gallai’s Conjecture for triangle-free planar graphs.     

Tutte’s 5-flow conjecture for highly cyclically connected cubic graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let ω(G) be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph G. Tutte’s conjecture is equivalent to its restriction to cubic graphs with ω≥2. We show that if a cubic graph G has no edge cut with fewer than edges that separates two odd cycles of a minimum 2-factor of G, then G has a nowhere-zero 5-flow. This implies that if a cubic graph G is cyclically n-edge connected and , then G has a nowhere-zero 5-flow.     

The Erdös-Sós conjecture for spiders

A classical result on extremal graph theory is the Erdös-Gallai theorem: if a graph on n vertices has more than edges, then it contains a path of k edges. Motivated by the result, Erdös and Sós conjectured that under the same condition, the graph should contain every tree of k edges. A spider is a rooted tree in which each vertex has degree one or two, except for the root. A leg of a spider is a path from the root to a vertex of degree one. Thus, a path is a spider of 1 or 2 legs. From the motivation, it is natural to consider spiders of 3 legs. In this paper, we prove that if a graph on n vertices has more than edges, then it contains every k-edge spider of 3 legs, and also, every k-edge spider with no leg of length more than 4, which strengthens a result of Wo?niak on spiders of diameter at most 4.     

A variant of the Erdős-Sós conjecture

A well-known conjecture of Erdős and Sós states that every graph with average degree exceeding $m-1$ contains every tree with $m$ edges as a subgraph. We propose a variant of this conjecture, which states that every graph of maximum degree exceeding $m$ and minimum degree at least $\lfloor 2m/3\rfloor$ contains every tree with $m$ edges. As evidence for our conjecture we show (a) for every $m$ there is a $g\left(m\right)$ such that the weakening of the conjecture obtained by replacing the first $m$ by $g\left(m\right)$ holds, and (b) there is a such that the weakening of the conjecture obtained by replacing $\lfloor 2m/3\rfloor$ by $\text{◂⋅▸}\left(1-\gamma \right)m$ holds.     

A note on the strength and minimum color sum of bipartite graphs

The strength of a graph G is the smallest integer s such that there exists a minimum sum coloring of G using integers {1,…,s}, only. For bipartite graphs of maximum degree Δ we show the following simple bound: s≤⌈Δ/2⌉+1. As a consequence, there exists a quadratic time algorithm for determining the strength and minimum color sum of bipartite graphs of maximum degree Δ≤4.     

An approximate version of Hadwiger's conjecture for claw‐free graphs

Hadwiger's conjecture states that every graph with chromatic number χ has a clique minor of size χ. In this paper we prove a weakened version of this conjecture for the class of claw‐free graphs (graphs that do not have a vertex with three pairwise nonadjacent neighbors). Our main result is that a claw‐free graph with chromatic number χ has a clique minor of size $\lceil\frac{2}{3}\chi\rceil$. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 259–278, 2010     

An approximate version of the Loebl-Komlós-Sós conjecture

Loebl, Komlós, and Sós conjectured that if at least half of the vertices of a graph G have degree at least some k∈N, then every tree with at most k edges is a subgraph of G. Our main result is an approximate version of this conjecture for large enough n=|V(G)|, assumed that n=O(k).Our result implies an asymptotic bound for the Ramsey number of trees. We prove that r(Tk,Tm)?k+m+o(k+m), as k+m→∞.     

Towards generalizing MacDougall’s conjecture on vertex-magic total labelings

MacDougall’s conjecture states that every regular graph of degree at least 2 has a vertex-magic total labeling (VMTL) with the lone exception of $2K_3$. Since there is enormous empirical evidence supporting this conjecture, it is reasonable to seek generalizations. Thus we ask the more general question: to what extent does the degree sequence of a graph determine the existence or nonexistence of a VMTL? We provide beginning steps towards answering this question, and related questions, by providing infinite families of degree sequences, and for each sequence, a graph with a VMTL and another graph without a VMTL.     

Vaught's conjecture for quite o-minimal theories

We study Vaught's problem for quite o-minimal theories. Quite o-minimal theories form a subclass of the class of weakly o-minimal theories preserving a series of properties of o-minimal theories. We investigate quite o-minimal theories having fewer than $2^{\omega }$ countable models and prove that the Exchange Principle for algebraic closure holds in any model of such a theory and also we prove binarity of these theories. The main result of the paper is that any quite o-minimal theory has either $2^{\omega }$ countable models or $6^a{3}^b$ countable models, where a and b are natural numbers.     

An application of Tutte’s Theorem to 1-factorization of regular graphs of high degree

A well known conjecture in graph theory states that every regular graph of even order 2n and degree λ(2n), where λ≥1/2, is 1-factorizable. Chetwynd and Hilton [A.G. Chetwynd, A.J.W. Hilton, 1-factorizing regular graphs of high degree — An improved bound, Discrete Math. 75 (1989) 103-112] and, independently, Niessen and Volkmann [T. Niessen, L. Volkmann, Class 1 conditions depending on the minimum degree and the number of vertices of maximum degree, J. Graph Theory (2) 14 (1990) 225-246] proved the above conjecture under the assumption that . Since these results were published no significant or even partial improvement has been made in terms of lowering the bound on λ. We shall obtain here a partial improvement on λ. Specifically, using the original Chetwynd-Hilton approach and Tutte’s 1-Factor Theorem, we show that the above bound can be improved to , apart (possibly) from two families of exceptional cases. We then show, under the stronger assumption that λ≥λ^∗≈0.785, that one of the two families of exceptional cases cannot occur.